Solved: Finding the Dimension of an Eigenspace In

Verifying Eigenvalues and Eigenvectors In Exercises 16, verify that i is an eigenvalue of A and that xi is a corresponding eigenvector. A = [ 2 0 0 2], 1 = 2, x1 = (1, 0) 2 = 2, x2 = (0, 1)

Verifying Eigenvalues and Eigenvectors In Exercises 16, verify that i is an eigenvalue of A and that xi is a corresponding eigenvector. A = [ 4 2 5 3], 1 = 1, x1 = (1, 1) 2 = 2, x2 = (5, 2)

Verifying Eigenvalues and Eigenvectors In Exercises 16, verify that i is an eigenvalue of A and that xi is a corresponding eigenvector. A = [ 2 0 0 3 1 0 1 2 3 ] , 1 = 2, x1 = (1, 0, 0) 2 = 1, x2 = (1, 1, 0) 3 = 3, x3 = (5, 1, 2)

Verifying Eigenvalues and Eigenvectors In Exercises 16, verify that i is an eigenvalue of A and that xi is a corresponding eigenvector. A = [ 2 2 1 2 1 2 3 6 0 ] , 1 = 5, x1 = (1, 2, 1) 2 = 3, x2 = (2, 1, 0) 3 = 3, x3 = (3, 0, 1)

Verifying Eigenvalues and Eigenvectors In Exercises 16, verify that i is an eigenvalue of A and that xi is a corresponding eigenvector. A = [ 0 0 1 1 0 0 0 1 0 ] , 1 = 1, x1 = (1, 1, 1)

Verifying Eigenvalues and Eigenvectors In Exercises 16, verify that i is an eigenvalue of A and that xi is a corresponding eigenvector. A = [ 4 0 0 1 2 0 3 1 3 ] , 1 = 4, x1 = (1, 0, 0) 2 = 2, x2 = (1, 2, 0) 3 = 3, x3 = (2, 1, 1)

Use A, i , and xi from Exercise 1 to show that (a) A(cx1) = 2(cx1) for any real number c. (b) A(cx2) = 2(cx2) for any real number c.

Use A, i , and xi from Exercise 4 to show that (a) A(cx1) = 5(cx1) for any real number c. (b) A(cx2) = 3(cx2) for any real number c. (c) A(cx3) = 3(cx3) for any real number c.

Determining Eigenvectors In Exercises 912, determine whether x is an eigenvector of A. A = [ 7 2 2 4] (a) x = (1, 2) (b) x = (2, 1) (c) x = (1, 2) (d) x = (1, 0)

Determining Eigenvectors In Exercises 912, determine whether x is an eigenvector of A. A = [ 3 5 10 2] (a) x = (4, 4) (b) x = (8, 4) (c) x = (4, 8) (d) x = (5, 3)

Determining Eigenvectors In Exercises 912, determine whether x is an eigenvector of A. A = [ 1 2 3 1 0 3 1 2 1 ] (a) x = (2, 4, 6) (b) x = (2, 0, 6) (c) x = (2, 2, 0) (d) x = (1, 0, 1)

Determining Eigenvectors In Exercises 912, determine whether x is an eigenvector of A. A = [ 1 0 1 0 2 2 5 4 9 ] (a) x = (1, 1, 0) (b) x = (5, 2, 1) (c) x = (0, 0, 0) (d) x = (26 3, 26 + 6, 3)

Finding Eigenspaces in R2 Geometrically In Exercises 13 and 14, use the method shown in Example 3 to find the eigenvalue(s) and corresponding eigenspace(s) of A. A = [ 1 0 0 1]

Finding Eigenspaces in R2 Geometrically In Exercises 13 and 14, use the method shown in Example 3 to find the eigenvalue(s) and corresponding eigenspace(s) of A. A = [ 1 0 k 1]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 6 2 3 1]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 1 2 4 8]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 1 2 2 1]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 2 1 4 1]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 1 1 2 3 2 1]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 1 4 1 2 1 4 0]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 2 0 0 2 3 1 3 2 2 ]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 3 0 0 2 0 2 1 2 0 ]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 1 2 6 2 5 6 2 2 3 ]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 3 3 1 2 4 2 3 9 5 ]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 0 4 0 3 4 0 5 10 4 ]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 1 2 3 2 3 2 13 2 9 2 5 2 10 8 ]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 2 0 0 0 0 2 0 0 0 0 3 4 0 0 1 0 ]

Characteristic Equation, Eigenvalues, and Eigenvectors In Exercises 1528, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. [ 5 1 0 0 0 4 0 0 0 0 1 0 0 0 3 4 ]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 4 2 5 3]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 2 3 3 6]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 1 2 1 3 1 3 1 3 ]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 1 2 1 2 1 2 1 2 ]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 2 1 1 4 0 4 2 1 5 ]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 1 1 1 2 0 1 1 1 2 ]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 3 1 3 0 1 2 1 6 0 5 1 4 4 ]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 1 2 2 1 0 1 5 0 5 1 4 3 ]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 1 2 3 4 1 2 3 4 2 4 6 8 3 6 9 12]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 1 4 0 0 1 4 0 0 0 0 1 2 0 0 1 2 ]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 1 0 2 0 0 1 0 2 1 0 2 0 1 1 2 2 ]

Finding Eigenvalues In Exercises 2940, use a software program or a graphing utility to find the eigenvalues of the matrix. [ 1 1 2 1 3 4 0 0 3 3 1 0 3 3 1 0

Eigenvalues of Triangular and Diagonal Matrices In Exercises 4144, find the eigenvalues of the triangular or diagonal matrix. [ 2 0 0 0 3 0 1 4 1 ]

Eigenvalues of Triangular and Diagonal Matrices In Exercises 4144, find the eigenvalues of the triangular or diagonal matrix. [ 5 3 4 0 7 2 0 0 3 ]

Eigenvalues of Triangular and Diagonal Matrices In Exercises 4144, find the eigenvalues of the triangular or diagonal matrix. [ 6 0 0 0 0 5 0 0 0 0 4 0 0 0 0 4 ]

Eigenvalues of Triangular and Diagonal Matrices In Exercises 4144, find the eigenvalues of the triangular or diagonal matrix. [ 1 2 0 0 0 0 5 4 0 0 0 0 0 0 0 0 0 3 4 ]

Eigenvalues and Eigenvectors of Linear Transformations In Exercises 4548, consider the linear transformation T: RnRn whose matrix A relative to the standard basis is given. Find (a) the eigenvalues of A, (b) a basis for each of the corresponding eigenspaces, and (c) the matrix A for T relative to the basis B, where B is made up of the basis vectors found in part (b). [ 2 1 2 5]

Eigenvalues and Eigenvectors of Linear Transformations In Exercises 4548, consider the linear transformation T: RnRn whose matrix A relative to the standard basis is given. Find (a) the eigenvalues of A, (b) a basis for each of the corresponding eigenspaces, and (c) the matrix A for T relative to the basis B, where B is made up of the basis vectors found in part (b). [ 8 1 16 2]

Eigenvalues and Eigenvectors of Linear Transformations In Exercises 4548, consider the linear transformation T: RnRn whose matrix A relative to the standard basis is given. Find (a) the eigenvalues of A, (b) a basis for each of the corresponding eigenspaces, and (c) the matrix A for T relative to the basis B, where B is made up of the basis vectors found in part (b). [ 0 1 0 2 3 0 1 1 1 ]

Eigenvalues and Eigenvectors of Linear Transformations In Exercises 4548, consider the linear transformation T: RnRn whose matrix A relative to the standard basis is given. Find (a) the eigenvalues of A, (b) a basis for each of the corresponding eigenspaces, and (c) the matrix A for T relative to the basis B, where B is made up of the basis vectors found in part (b). [ 3 2 5 1 4 5 4 0 6 ]

Cayley-Hamilton Theorem In Exercises 4952, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of A = [ 1 2 3 5] is 2 6 + 11 = 0, and by the theorem you have A2 6A + 11I2 = O. A = [ 5 7 0 3]

Cayley-Hamilton Theorem In Exercises 4952, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of A = [ 1 2 3 5] is 2 6 + 11 = 0, and by the theorem you have A2 6A + 11I2 = O. A = [ 6 1 1 5]

Cayley-Hamilton Theorem In Exercises 4952, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of A = [ 1 2 3 5] is 2 6 + 11 = 0, and by the theorem you have A2 6A + 11I2 = O. A = [ 1 0 2 0 3 0 4 1 1 ]

Cayley-Hamilton Theorem In Exercises 4952, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of A = [ 1 2 3 5] is 2 6 + 11 = 0, and by the theorem you have A2 6A + 11I2 = O. A = [ 3 1 0 1 3 4 0 2 3 ]

Perform each computational check on the eigenvalues found in Exercises 1527 odd. (a) The sum of the n eigenvalues equals the trace of the matrix. (Recall that the trace of a matrix is the sum of the main diagonal entries of the matrix.) (b) The product of the n eigenvalues equals A. (When is an eigenvalue of multiplicity k, remember to use it k times in the sum or product of these checks.)

Perform each computational check on the eigenvalues found in Exercises 1628 even. (a) The sum of the n eigenvalues equals the trace of the matrix. (Recall that the trace of a matrix is the sum of the main diagonal entries of the matrix.) (b) The product of the n eigenvalues equals A. (When is an eigenvalue of multiplicity k, remember to use it k times in the sum or product of these checks.)

Show that if A is an n n matrix whose ith row is identical to the ith row of I, then 1 is an eigenvalue of A.

Proof Prove that = 0 is an eigenvalue of A if and only if A is singular.

Proof For an invertible matrix A, prove that A and A1 have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A1?

Proof Prove that A and AT have the same eigenvalues. Are the eigenspaces the same?

Proof Prove that the constant term of the characteristic polynomial is A.

Define T: R2R2 by T(v) = projuv where u is a fixed vector in R2. Show that the eigenvalues of A (the standard matrix of T ) are 0 and 1.

Guided Proof Prove that a triangular matrix is nonsingular if and only if its eigenvalues are real and nonzero. Getting Started: This is an if and only if statement, so you must prove that the statement is true in both directions. Review Theorems 3.2 and 3.7. (i) To prove the statement in one direction, assume that the triangular matrix A is nonsingular. Use your knowledge of nonsingular and triangular matrices and determinants to conclude that the entries on the main diagonal of A are nonzero. (ii) A is triangular, so use Theorem 7.3 and part (i) to conclude that the eigenvalues are real and nonzero. (iii) To prove the statement in the other direction, assume that the eigenvalues of the triangular matrix A are real and nonzero. Repeat parts (i) and (ii) in reverse order to prove that A is nonsingular.

Guided Proof Prove that if A2 = O, then 0 is the only eigenvalue of A. Getting Started: You need to show that if there exists a nonzero vector x and a real number such that Ax = x, then if A2 = O, must be zero. (i) A2 = A A, so you can write A2x as A(Ax). (ii) Use the fact that Ax = x and the properties of matrix multiplication to show that A2x = 2x. (iii) A2 is a zero matrix, so you can conclude that must be zero

Proof Prove that the multiplicity of an eigenvalue is greater than or equal to the dimension of its eigenspace.

CAPSTONE An n n matrix A has the characteristic equation I A = ( + 2)( 1)( 3)2 = 0. (a) What are the eigenvalues of A? (b) What is the order of A? Explain. (c) Is I A singular? Explain. (d) Is A singular? Explain. (Hint: Use the result of Exercise 56.)

When the eigenvalues of A = [ a 0 b d] are 1 = 0 and 2 = 1, what are the possible values of a and d?

Show that A = [ 0 1 1 0] has no real eigenvalues.

True or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The scalar is an eigenvalue of an n n matrix A when there exists a vector x such that Ax = x. (b) To find the eigenvalue(s) of an n n matrix A, you can solve the characteristic equation det(I A) = 0.

True or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) Geometrically, if is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to , then multiplying x by A produces a vector x parallel to x. (b) If A is an n n matrix with an eigenvalue , then the set of all eigenvectors of is a subspace of Rn.

Finding the Dimension of an Eigenspace In Exercises 6972, find the dimension of the eigenspace corresponding to the eigenvalue = 3. . A = [ 3 0 0 0 3 0 0 0 3 ]

Finding the Dimension of an Eigenspace In Exercises 6972, find the dimension of the eigenspace corresponding to the eigenvalue = 3. A = [ 3 0 0 1 3 0 0 0 3 ]

Finding the Dimension of an Eigenspace In Exercises 6972, find the dimension of the eigenspace corresponding to the eigenvalue = 3. A = [ 3 0 0 1 3 0 0 1 3 ]

Finding the Dimension of an Eigenspace In Exercises 6972, find the dimension of the eigenspace corresponding to the eigenvalue = 3. A = [ 3 0 0 1 3 0 1 1 3 ]

Calculus Let T: C[0, 1]C[0, 1] be the linear transformation T(f) = f. Show that = 1 is an eigenvalue of T with corresponding eigenvector f(x) = ex .

Calculus For the linear transformation in Exercise 73, find the eigenvalue corresponding to the eigenvector f(x) = e2x .

Define T: P2P2 by T(a0 + a1x + a2x2) = (3a1 + 5a2) + (4a0 + 4a1 10a2)x + 4a2x2. Find the eigenvalues and the eigenvectors of T relative to the standard basis {1, x, x2}.

Define T: P2P2 by T(a0 + a1x + a2x2) = (2a0 + a1 a2) + (a1 + 2a2)x a2x 2. Find the eigenvalues and eigenvectors of T relative to the standard basis {1, x, x2}.

Define T: M2,2M2,2 by T([ a c b d]) = [ a c + d 2a + 2c 2d b + d 2b + 2d] Find the eigenvalues and eigenvectors of T relative to the standard basis B = {[ 1 0 0 0], [ 0 0 1 0], [ 0 1 0 0], [ 0 0 0 1]}.

Find all values of the angle for which the matrix A = [ cos sin sin cos ] has real eigenvalues. Interpret your answer geometrically

What are the possible eigenvalues of an idempotent matrix? (Recall that a square matrix A is idempotent when A2 = A.)

What are the possible eigenvalues of a nilpotent matrix? (Recall that a square matrix A is nilpotent when there exists a positive integer k such that Ak = 0.)

Proof Let A be an n n matrix such that the sum of the entries in each row is a fixed constant r. Prove that r is an eigenvalue of A. Illustrate this result with an example.

In Exercises 1–6, verify that \(\lambda_{i}\) is an eigenvalue of A and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. \(A=\left[\begin{array}{rr} 2 & 0 \\ 0 & -2 \end{array}\right], \begin{array}{l} \lambda_{1}=2, \mathbf{x}_{1}=(1,0) \\ \lambda_{2}=-2, \mathbf{x}_{2}=(0,1) \end{array}\) Text Transcription: lambda_i x_i A=[2 0 0 -2], lambda_1=2, x_1=(1,0) lambda_2=-2, x_2=(0,1)

In Exercises 1–6, verify that \(\lambda_{i}\) is an eigenvalue of A and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. \(A=\left[\begin{array}{ll} 4 & -5 \\ 2 & -3 \end{array}\right], \begin{array}{l} \lambda_{1}=-1, \mathbf{x}_{1}=(1,1) \\ \lambda_{2}=2, \mathbf{x}_{2}=(5,2) \end{array}\) Text Transcription: lambda_i x_i A=[4 -5 2 -3], lambda_1=-1, x_1=(1,1) lambda_2=2, x_2=(5,2)

In Exercises 1–6, verify that \(\lambda_{i}\) is an eigenvalue of A and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. \(A=\left[\begin{array}{rrr} 2 & 3 & 1 \\ 0 & -1 & 2 \\ 0 & 0 & 3 \end{array}\right], \begin{array}{l} \lambda_{1}=2, \mathbf{x}_{1}=(1,0,0) \\ \lambda_{2}=-1, \mathbf{x}_{2}=(1,-1,0) \\ \lambda_{3}=3, \mathbf{x}_{3}=(5,1,2) \end{array}\) Text Transcription: lambda_i x_i A=[2 3 1 0 -1 2 0 0 3], lambda_1=2, x_1=(1,0,0) lambda_2=-1, x_2=(1,-1,0) lambda_3=3, x_3=(5,1,2)

In Exercises 1–6, verify that \(\lambda_{i}\) is an eigenvalue of A and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. \(A=\left[\begin{array}{rrr} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{array}\right], \begin{array}{l} \lambda_{1}=5, \mathbf{x}_{1}=(1,2,-1) \\ \lambda_{2}=-3, \mathbf{x}_{2}=(-2,1,0) \\ \lambda_{3}=-3, \mathbf{x}_{3}=(3,0,1) \end{array}\) Text Transcription: lambda_i x_i A=[-2 2 -3 2 1 -6 -1 -2 0], lambda_1=5, x_1=(1,2,-1) lambda_2=-3, x_2=(-2,1,0) lambda_3=-3, x_3=(3,0,1)

In Exercises 1–6, verify that \(\lambda_{i}\) is an eigenvalue of A and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. \(A=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right], \lambda_{1}=1, \mathbf{x}_{1}=(1,1,1)\) Text Transcription: lambda_i x_i A=[0 1 0 0 0 1 1 0 0], lambda_1=1, x_1=(1,1,1)

In Exercises 1–6, verify that \(\lambda_{i}\) is an eigenvalue of A and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. \(A=\left[\begin{array}{rrr} 4 & -1 & 3 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \end{array}\right], \begin{array}{l} \lambda_{1}=4, \mathbf{x}_{1}=(1,0,0) \\ \lambda_{2}=2, \mathbf{x}_{2}=(1,2,0) \\ \lambda_{3}=3, \mathbf{x}_{3}=(-2,1,1) \end{array}\) Text Transcription: lambda_i x_i A=[4 -1 3 0 2 1 0 0 3], lambda_1=4, x_1=(1,0,0) lambda_2=2, x_2=(1,2,0) lambda_3=3, x_3=(-2,1,1)

Use A, \(\lambda_{i}, \text { and } \mathbf{x}_{i}\) from Exercise 1 to show that (a) \(A\left(c \mathbf{x}_{1}\right)=2\left(c \mathbf{x}_{1}\right)\) for any real number c. (b) \(A\left(c \mathbf{x}_{2}\right)=-2\left(c \mathbf{x}_{2}\right)\) for any real number c. Text Transcription: lambda_i, and x_i A (cx_1) = 2(cx_1) A (cx_2) = -2(cx_2)

Use A, (\lambda_{i}, \text { and } \mathbf{x}_{i}\) from Exercise 4 to show that (a) \(A\left(c \mathbf{x}_{1}\right)=5\left(c \mathbf{x}_{1}\right)\) for any real number c. (b) \(A\left(c \mathbf{x}_{2}\right)=-3\left(c \mathbf{x}_{2}\right)\) for any real number c. (c) \(A\left(c \mathbf{x}_{3}\right)=-3\left(c \mathbf{x}_{3}\right)\) for any real number c. Text Transcription: lambda_i, and x_i A (cx_1)= 5 (cx_1) A (cx_2)=-3 (cx_2) A (cx_3)=-3 (cx_3)

In Exercises 9–12, determine whether x is an eigenvector of A. \(A=\left[\begin{array}{ll} 7 & 2 \\ 2 & 4 \end{array}\right]\) (a) x = (1, 2) (b) x = (2, 1) (c) x = (1, ?2) (d) x = (?1, 0) Text Transcription: A=[7 2 2 4]

In Exercises 9–12, determine whether x is an eigenvector of A. \(A=\left[\begin{array}{rr} -3 & 10 \\ 5 & 2 \end{array}\right]\) (a) x = (4, 4) (b) x = (?8, 4) (c) x = (?4, 8) (d) x = (5, ?3) Text Transcription: A=[-3 10 5 2]

In Exercises 9–12, determine whether x is an eigenvector of A. \(A=\left[\begin{array}{rrr} -1 & -1 & 1 \\ -2 & 0 & -2 \\ 3 & -3 & 1 \end{array}\right]\) (a) x = (2, ?4, 6) (b) x = (2, 0, 6) (c) x = (2, 2, 0) (d) x = (?1, 0, 1) Text Transcription: A=[-1 -1 1 -2 0 -2 3 -3 1]

In Exercises 9–12, determine whether x is an eigenvector of A. \(A=\left[\begin{array}{rrr} 1 & 0 & 5 \\ 0 & -2 & 4 \\ 1 & -2 & 9 \end{array}\right]\) (a) x = (1, 1, 0) (b) x = (?5, 2, 1) (c) x = (0, 0, 0) (d) \(x=(2 \sqrt{6}-3,-2 \sqrt{6}+6,3)\) Text Transcription: A=[1 0 5 0 -2 4 1 -2 9] x = (2 sqrt 6-3,-2 sqrt 6+6,3)

In Exercises 13 and 14, use the method shown in Example 3 to find the eigenvalue(s) and corresponding eigenspace(s) of A. \(A=\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right]\) Text Transcription: A=[1 0 0 -1]

In Exercises 13 and 14, use the method shown in Example 3 to find the eigenvalue(s) and corresponding eigenspace(s) of A. \(A=\left[\begin{array}{ll} 1 & k \\ 0 & 1 \end{array}\right]\) Text Transcription: A=[1 k 0 1]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{rr} 6 & -3 \\ -2 & 1 \end{array}\right]\) Text Transcription: [6 -3 -2 1]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{rr} 1 & -4 \\ -2 & 8 \end{array}\right]\) Text Transcription: [1 -4 -2 8]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right]\) Text Transcription: [1 2 2 1]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{rr} -2 & 4 \\ 1 & 1 \end{array}\right]\) Text Transcription: [-2 4 1 1]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{ll} 1 & -\frac{3}{2} \\ \frac{1}{2} & -1 \end{array}\right]\) Text Transcription: [1 -3/2 1/2 -1]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{ll} \frac{1}{4} & \frac{1}{4} \\ \frac{1}{2} & 0 \end{array}\right]\) Text Transcription: [1/4 1/4 1/2 0]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{rrr} 2 & -2 & 3 \\ 0 & 3 & -2 \\ 0 & -1 & 2 \end{array}\right]\) Text Transcription: [2 -2 3 0 3 -2 0 -1 2]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\\left[\begin{array}{lll} 3 & 2 & 1 \\ 0 & 0 & 2 \\ 0 & 2 & 0 \end{array}\right]\) Text Transcription: [3 2 1 0 0 2 0 2 0]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{rrr} 1 & 2 & -2 \\ -2 & 5 & -2 \\ -6 & 6 & -3 \end{array}\right]\) Text Transcription: [1 2 -2 -2 5 -2 -6 6 -3]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{rrr} 3 & 2 & -3 \\ -3 & -4 & 9 \\ -1 & -2 & 5 \end{array}\right]\) Text Transcription: [3 2 -3 -3 -4 9 -1 -2 5]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{rrr} 0 & -3 & 5 \\ -4 & 4 & -10 \\ 0 & 0 & 4 \end{array}\right]\) Text Transcription: [0 -3 5 -4 4 -10 0 0 4]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{rrr} 1 & -\frac{3}{2} & \frac{5}{2} \\ -2 & \frac{13}{2} & -10 \\ \frac{3}{2} & -\frac{9}{2} & 8 \end{array}\right]\) Text Transcription: [1 -3/2 5/2 -2 13/2 -10 3/2 -9/2 8]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{llll} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 4 & 0 \end{array}\right]\) Text Transcription: [2 0 0 0 0 2 0 0 0 0 3 1 0 0 4 0]

In Exercises 15–28, find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. \(\left[\begin{array}{llll} 5 & 0 & 0 & 0 \\ 1 & 4 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 4 \end{array}\right]\) Text Transcription: [5 0 0 0 1 4 0 0 0 0 1 3 0 0 0 4]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{ll} -4 & 5 \\ -2 & 3 \end{array}\right]\) Text Transcription: [-4 5 -2 3]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rr} 2 & 3 \\ 3 & -6 \end{array}\right]\) Text Transcription: [2 3 3 -6]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rr} \frac{1}{2} & \frac{1}{3} \\ -\frac{1}{3} & -\frac{1}{3} \end{array}\right]\) Text Transcription: [1/2 1/3 -1/3 -13]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rr} \frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{2} & -\frac{1}{2} \end{array}\right]\) Text Transcription: [1/2 -1/2 -1/2 -1/2]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rrr} 2 & 4 & 2 \\ 1 & 0 & 1 \\ 1 & -4 & 5 \end{array}\right]\) Text Transcription: [2 4 2 1 0 1 1 -4 5]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rrr} 1 & 2 & -1 \\ 1 & 0 & 1 \\ 1 & -1 & 2 \end{array}\right]\) Text Transcription: [1 2 -1 1 0 1 1 -1 2]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rrr} 3 & -\frac{1}{2} & 5 \\ -\frac{1}{3} & -\frac{1}{6} & -\frac{1}{4} \\ 0 & 0 & 4 \end{array}\right]\) Text Transcription: [3 -1/2 5 -1/3 -1/6 -1/4 0 0 4]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rrr} \frac{1}{2} & 0 & 5 \\ -2 & \frac{1}{5} & \frac{1}{4} \\ 1 & 0 & 3 \end{array}\right]\) Text Transcription: [1/2 0 5 -2 1/5 1/4 1 0 3]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rrrr} 1 & 1 & 2 & 3 \\ 2 & 2 & 4 & 6 \\ 3 & 3 & 6 & 9 \\ 4 & 4 & 8 & 12 \end{array}\right]\) Text Transcription: [1 1 2 3 2 2 4 6 3 3 6 9 4 4 8 12]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{llll} 1 & 1 & 0 & 0 \\ 4 & 4 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 2 & 2 \end{array}\right]\) Text Transcription: [1 1 0 0 4 4 0 0 0 0 1 1 0 0 2 2]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rrrr} 1 & 0 & -1 & 1 \\ 0 & 1 & 0 & 1 \\ -2 & 0 & 2 & -2 \\ 0 & 2 & 0 & 2 \end{array}\right]\) Text Transcription: [1 0 -1 1 0 1 0 1 -2 0 2 -2 0 2 0 2]

In Exercises 29–40, use a software program or a graphing utility to find the eigenvalues of the matrix. \(\left[\begin{array}{rrrr} 1 & -3 & 3 & 3 \\ -1 & 4 & -3 & -3 \\ -2 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right]\) Text Transcription: [1 -3 3 3 -1 4 -3 -3 -2 0 1 1 1 0 0 0]

In Exercises 41–44, find the eigenvalues of the triangular or diagonal matrix. \(\left[\begin{array}{lll} 2 & 0 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 1 \end{array}\right]\) Text Transcription: [2 0 1 0 3 4 0 0 1]

In Exercises 41–44, find the eigenvalues of the triangular or diagonal matrix. \(\left[\begin{array}{rrr} -5 & 0 & 0 \\ 3 & 7 & 0 \\ 4 & -2 & 3 \end{array}\right]\) Text Transcription: [-5 0 0 3 7 0 4 -2 3]

In Exercises 41–44, find the eigenvalues of the triangular or diagonal matrix. \(\left[\begin{array}{rrrr} -6 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & -4 & 0 \\ 0 & 0 & 0 & -4 \end{array}\right]\) Text Transcription: [-6 0 0 0 0 5 0 0 0 0 -4 0 0 0 0 -4]

In Exercises 41–44, find the eigenvalues of the triangular or diagonal matrix. \(\left[\begin{array}{cccc} \frac{1}{2} & 0 & 0 & 0 \\ 0 & \frac{5}{4} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{4} \end{array}\right]\) Text Transcription: [1/2 0 0 0 0 5/4 0 0 0 0 0 0 0 0 0 3/4]

In Exercises 45–48, consider the linear transformation \(T: R^{n} \rightarrow R^{n}\) whose matrix A relative to the standard basis is given. Find (a) the eigenvalues of A, (b) a basis for each of the corresponding eigenspaces, and (c) the matrix A’ for T relative to the basis B’, where B’ is made up of the basis vectors found in part (b). \(\left[\begin{array}{rr} 2 & -2 \\ 1 & 5 \end{array}\right]\) Text Transcription: T: R^n rightarrow R^n [2 -2 1 5]

In Exercises 45–48, consider the linear transformation \(T: R^{n} \rightarrow R^{n}\) whose matrix A relative to the standard basis is given. Find (a) the eigenvalues of A, (b) a basis for each of the corresponding eigenspaces, and (c) the matrix A’ for T relative to the basis B’, where B’ is made up of the basis vectors found in part (b). \(\left[\begin{array}{rr} -8 & 16 \\ 1 & -2 \end{array}\right]\) Text Transcription: T: R^n rightarrow R^n [-8 16 1 -2]

In Exercises 45–48, consider the linear transformation \(T: R^{n} \rightarrow R^{n}\) whose matrix A relative to the standard basis is given. Find (a) the eigenvalues of A, (b) a basis for each of the corresponding eigenspaces, and (c) the matrix A’ for T relative to the basis B’, where B’ is made up of the basis vectors found in part (b). \(\left[\begin{array}{rrr} 0 & 2 & -1 \\ -1 & 3 & 1 \\ 0 & 0 & -1 \end{array}\right]\) Text Transcription: T: R^n rightarrow R^n [0 2 -1 -1 3 1 0 0 -1]

In Exercises 45–48, consider the linear transformation \(T: R^{n} \rightarrow R^{n}\) whose matrix A relative to the standard basis is given. Find (a) the eigenvalues of A, (b) a basis for each of the corresponding eigenspaces, and (c) the matrix A’ for T relative to the basis B’, where B’ is made up of the basis vectors found in part (b). \(\left[\begin{array}{lll} 3 & 1 & 4 \\ 2 & 4 & 0 \\ 5 & 5 & 6 \end{array}\right]\) Text Transcription: T: R^n rightarrow R^n [3 1 4 2 4 0 5 5 6]

In Exercises 49–52, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of \(A=\left[\begin{array}{rr} 1 & -3 \\ 2 & 5 \end{array}\right]\) is \(\lambda^{2}-6 \lambda+11=0\), and by the theorem you have \(A^{2}-6 A+11 I_{0}=0\). \(A=\left[\begin{array}{rr} 5 & 0 \\ -7 & 3 \end{array}\right]\) Text Transcription: A=[1 -3 2 5] lambda^2-6 lambda+11=0 A^2-6 A+11 I_0=0 A=[5 0 -7 3]

In Exercises 49–52, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of \(A=\left[\begin{array}{rr} 1 & -3 \\ 2 & 5 \end{array}\right]\) is \(\lambda^{2}-6 \lambda+11=0\), and by the theorem you have \(A^{2}-6 A+11 I_{0}=0\). \(A=\left[\begin{array}{rr} 6 & -1 \\ 1 & 5 \end{array}\right]\) Text Transcription: A=[1 -3 2 5] lambda^2-6 lambda+11=0 A^2-6 A+11 I_0=0 A=[6 -1 1 5]

In Exercises 49–52, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of \(A=\left[\begin{array}{rr} 1 & -3 \\ 2 & 5 \end{array}\right]\) is \(\lambda^{2}-6 \lambda+11=0\), and by the theorem you have \(A^{2}-6 A+11 I_{0}=0\). \(A=\left[\begin{array}{rrr} 1 & 0 & -4 \\ 0 & 3 & 1 \\ 2 & 0 & 1 \end{array}\right]\) Text Transcription: A=[1 -3 2 5] lambda^2-6 lambda+11=0 A^2-6 A+11 I_0=0 A=[1 0 -4 0 3 1 2 0 1]

In Exercises 49–52, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of \(A=\left[\begin{array}{rr} 1 & -3 \\ 2 & 5 \end{array}\right]\) is \(\lambda^{2}-6 \lambda+11=0\), and by the theorem you have \(A^{2}-6 A+11 I_{0}=0\). \(A=\left[\begin{array}{rrr} -3 & 1 & 0 \\ -1 & 3 & 2 \\ 0 & 4 & 3 \end{array}\right]\) Text Transcription: A=[1 -3 2 5] lambda^2-6 lambda+11=0 A^2-6 A+11 I_0=0 A=[-3 1 0 -1 3 2 0 4 3]

Perform each computational check on the eigenvalues found in Exercises 15–27 odd. (a) The sum of the n eigenvalues equals the trace of the matrix. (Recall that the trace of a matrix is the sum of the main diagonal entries of the matrix.) (b) The product of the n eigenvalues equals |A|. (When \(\lambda\) is an eigenvalue of multiplicity k, remember to use it k times in the sum or product of these checks.) Text Transcription: lambda

Perform each computational check on the eigenvalues found in Exercises 16–28 even. (a) The sum of the n eigenvalues equals the trace of the matrix. (Recall that the trace of a matrix is the sum of the main diagonal entries of the matrix.) (b) The product of the n eigenvalues equals |A|. (When \(\lambda\) is an eigenvalue of multiplicity k, remember to use it k times in the sum or product of these checks.) Text Transcription: lambda

Show that if A is an \(n \times n\) matrix whose ith row is identical to the ith row of I, then 1 is an eigenvalue of A. Text Transcription: n times n

Prove that \(\lambda=0\) is an eigenvalue of A if and only if A is singular. Text Transcription: lambda=0

For an invertible matrix A, prove that A and \(A^{-1}\) have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of \(A^{-1}\)? Text Transcription: A^-1

Prove that A and \(A^{T}\) have the same eigenvalues. Are the eigenspaces the same? Text Transcription: A^T

Prove that the constant term of the characteristic polynomial is \(\pm|A|\). Text Transcription: pm|A|

Define T: R2R2 by T(v) = projuv where u is a fixed vector in R2. Show that the eigenvalues of A (the standard matrix of T ) are 0 and 1.

Prove that a triangular matrix is nonsingular if and only if its eigenvalues are real and nonzero. Getting Started: This is an “if and only if” statement, so you must prove that the statement is true in both directions. Review Theorems 3.2 and 3.7. (i) To prove the statement in one direction, assume that the triangular matrix A is nonsingular. Use your knowledge of nonsingular and triangular matrices and determinants to conclude that the entries on the main diagonal of A are nonzero. (ii) A is triangular, so use Theorem 7.3 and part (i) to conclude that the eigenvalues are real and nonzero. (iii) To prove the statement in the other direction, assume that the eigenvalues of the triangular matrix A are real and nonzero. Repeat parts (i) and (ii) in reverse order to prove that A is nonsingular.

Guided Proof Prove that if A2 = O, then 0 is the only eigenvalue of A. Getting Started: You need to show that if there exists a nonzero vector x and a real number such that Ax = x, then if A2 = O, must be zero. (i) A2 = A A, so you can write A2x as A(Ax). (ii) Use the fact that Ax = x and the properties of matrix multiplication to show that A2x = 2x. (iii) A2 is a zero matrix, so you can conclude that must be zero

Proof Prove that the multiplicity of an eigenvalue is greater than or equal to the dimension of its eigenspace.

An \(n \times n\) matrix A has the characteristic equation \(|\lambda I-A|=(\lambda+2)(\lambda-1)(\lambda-3)^{2}=0\) (a) What are the eigenvalues of A? (b) What is the order of A? Explain. (c) Is \(\lambda I-A\) singular? Explain. (d) Is A singular? Explain. (Hint: Use the result of Exercise 56.) Text Transcription: n times n |lambda I-A|=(lambda+2)(lambda-1)(lambda-3)^2=0 lambda I-A

When the eigenvalues of \(A=\left[\begin{array}{ll} a & b \\ 0 & d \end{array}\right]\) are \(\lambda_{1}=0 \text { and } \lambda_{2}=1\), what are the possible values of a and d? Text Transcription: A=[a b 0 d] lambda_1=0 and lambda_2=1

Show that \(A=\left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right]\) has no real eigenvalues. Text Transcription: A=[0 1 -1 0]

In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The scalar \(\lambda\) is an eigenvalue of an \(n \times n\) matrix A when there exists a vector x such that \(A \mathbf{x}=\lambda \mathbf{x}\). (b) To find the eigenvalue(s) of an \(n \times n\) matrix A, you can solve the characteristic equation \(\operatorname{det}(\lambda I-A)=0\). Text Transcription: lambda n times n Ax=lambda x det(lambda I-A)=0

In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) Geometrically, if \(\lambda\) is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to \(\lambda\), then multiplying x by A produces a vector \(\lambda \mathbf{x}\) parallel to x. (b) If A is an \(n \times n\) matrix with an eigenvalue \(\lambda\), then the set of all eigenvectors of \(\lambda\) is a subspace of \(R^{n}\). Text Transcription: lambda lambda x n times n R^n

In Exercises 69–72, find the dimension of the eigenspace corresponding to the eigenvalue \(\lambda=3\). \(A=\left[\begin{array}{lll} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}\right]\) Text Transcription: lambda=3 A=[3 0 0 0 3 0 0 0 3]

In Exercises 69–72, find the dimension of the eigenspace corresponding to the eigenvalue \(\lambda=3\). \(A=\left[\begin{array}{lll} 3 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}\right]\) Text Transcription: lambda=3 A=[3 1 0 0 3 0 0 0 3]

In Exercises 69–72, find the dimension of the eigenspace corresponding to the eigenvalue \(\lambda=3\). \(A=\left[\begin{array}{lll} 3 & 1 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 3 \end{array}\right]\) Text Transcription: lambda=3 A=[3 1 0 0 3 1 0 0 3]

In Exercises 69–72, find the dimension of the eigenspace corresponding to the eigenvalue \(\lambda=3\). \(A=\left[\begin{array}{lll} 3 & 1 & 1 \\ 0 & 3 & 1 \\ 0 & 0 & 3 \end{array}\right]\) Text Transcription: lambda=3 A=[3 1 1 0 3 1 0 0 3]

Let \(T: C^{\prime}[0,1] \rightarrow C[0,1]\) be the linear transformation T( f ) = f’ . Show that \(\lambda=1\) is an eigenvalue of T with corresponding eigenvector \(f(x)=e^{x}\). Text Transcription: T: C^prime[0,1] rightarrow C[0,1] lambda=1 f(x)=e^x

For the linear transformation in Exercise 73, find the eigenvalue corresponding to the eigenvector \(f(x)=e^{-2 x}\). Text Transcription: f(x)=e^-2x

Define \(T: P_{2} \rightarrow P_{2}\) by \(\begin{aligned} T\left(a_{0}+a_{1} x+a_{2} x^{2}\right)=&\left(-3 a_{1}+5 a_{2}\right)+\\ &\left(-4 a_{0}+4 a_{1}-10 a_{2}\right) x+4 a_{2} x^{2} \end{aligned}\). Find the eigenvalues and the eigenvectors of T relative to the standard basis \(\left\{1, x, x^{2}\right\}\). Text Transcription: T: P_2 rightarrow P_2 T(a_0+a_1x+a_2x^2)=(-3a_1+5a_2)+ (-4a_0+4a_1-10a_2)x+4a_2 x^2 {1, x, x^2}

Define \(T: P_{2} \rightarrow P_{2}\) by \(\begin{aligned} T\left(a_{0}+a_{1} x+a_{2} x^{2}\right)=&\left(2 a_{0}+a_{1}-a_{2}\right)+\\ &\left(-a_{1}+2 a_{2}\right) x-a_{2} x^{2} \end{aligned}\). Find the eigenvalues and the eigenvectors of T relative to the standard basis \(\left\{1, x, x^{2}\right\}\). Text Transcription: T: P_2 rightarrow P_2 T(a_0+a_1x+a_2x^2)=(2a_0+a_1-a_2)+ (-a_1+2 a_2) x-a_2x^2 {1, x, x^2}

Define \(T: M_{2,2} \rightarrow M_{2,2}\) by \(T\left(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\right)=\left[\begin{array}{rr} a-c+d & b+d \\ -2 a+2 c-2 d & 2 b+2 d \end{array}\right]\). Find the eigenvalues and eigenvectors of T relative to the standard basis \(B=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\}\) Text Transcription: T: M_2,2 rightarrow M_2,2 T (a b c d)=[a-c+d b+d -2a+2c-2d 2b+2d] B={[1 0 0 0], [0 1 0 0], [0 0 1 0], [0 0 0 1]}

Find all values of the angle \(\theta\) for which the matrix \(A=\left[\begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\) has real eigenvalues. Interpret your answer geometrically. Text Transcription: theta A=[cos theta -sin theta sin theta cos theta]

What are the possible eigenvalues of an idempotent matrix? (Recall that a square matrix A is idempotent when \(A^{2}=A\).) Text Transcription: A^2=A

What are the possible eigenvalues of a nilpotent matrix? (Recall that a square matrix A is nilpotent when there exists a positive integer k such that \(A^{k}=0\).) Text Transcription: A^k=0

Proof Let A be an n n matrix such that the sum of the entries in each row is a fixed constant r. Prove that r is an eigenvalue of A. Illustrate this result with an example.