 5.2.1: In Exercises 14, show that A and B are not similar matrices.A = 1 1...
 5.2.2: In Exercises 14, show that A and B are not similar matrices.A = 4 1...
 5.2.3: In Exercises 14, show that A and B are not similar matrices.A = 123...
 5.2.4: In Exercises 14, show that A and B are not similar matrices.A = 101...
 5.2.5: In Exercises 58, find a matrix P that diagonalizes A, and check you...
 5.2.6: In Exercises 58, find a matrix P that diagonalizes A, and check you...
 5.2.7: In Exercises 58, find a matrix P that diagonalizes A, and check you...
 5.2.8: In Exercises 58, find a matrix P that diagonalizes A, and check you...
 5.2.9: Let A = 401 232 104 (a) Find the eigenvalues of A. (b) For each eig...
 5.2.10: Follow the directions in Exercise 9 for the matrix 300 020 012
 5.2.11: In Exercises 1114, find the geometric and algebraic multiplicity of...
 5.2.12: In Exercises 1114, find the geometric and algebraic multiplicity of...
 5.2.13: In Exercises 1114, find the geometric and algebraic multiplicity of...
 5.2.14: In Exercises 1114, find the geometric and algebraic multiplicity of...
 5.2.15: In each part of Exercises 1516, the characteristic equation of a ma...
 5.2.16: In each part of Exercises 1516, the characteristic equation of a ma...
 5.2.17: In Exercises 1718, use the method of Example 6 to compute the matri...
 5.2.18: In Exercises 1718, use the method of Example 6 to compute the matri...
 5.2.19: Let A = 1 7 1 010 0 15 2 and P = 111 001 105 Confirm that P diagona...
 5.2.20: Let A = 1 2 8 0 1 0 0 0 1 and P = 1 4 1 100 010 Confirm that P diag...
 5.2.21: Let A = 1 2 8 0 1 0 0 0 1 and P = 1 4 1 100 010 Confirm that P diag...
 5.2.22: Show that the matrices A = 111 111 111 and B = 300 000 000 are simi...
 5.2.23: We know from Table 1 that similar matrices have the same rank. Show...
 5.2.24: We know from Table 1 that similar matrices have the same eigenvalue...
 5.2.25: If A, B, and C are n n matrices such that A is similar to B and B i...
 5.2.26: (a) Is it possible for an n n matrix to be similar to itself ? Just...
 5.2.27: Suppose that the characteristic polynomial of some matrix A is foun...
 5.2.28: Let A = a b c d Show that (a) A is diagonalizable if (a d)2 + 4bc >...
 5.2.29: In the case where the matrix A in Exercise 28 is diagonalizable, fi...
 5.2.30: In Exercises 3033, find the standard matrix A for the given linear ...
 5.2.31: In Exercises 3033, find the standard matrix A for the given linear ...
 5.2.32: In Exercises 3033, find the standard matrix A for the given linear ...
 5.2.33: In Exercises 3033, find the standard matrix A for the given linear ...
 5.2.34: If P is a fixed n n matrix, then the similarity transformation AP 1...
 5.2.35: Prove that similar matrices have the same rank and nullity.
 5.2.36: Prove that similar matrices have the same trace.
 5.2.37: Prove that if A is diagonalizable, then so is Ak for every positive...
 5.2.38: We know from Table 1 that similar matrices, A and B, have the same ...
 5.2.39: Let A be an n n matrix, and let q(A) be the matrix q(A) = anAn + an...
 5.2.40: Prove that if A is a diagonalizable matrix, then the rank of A is t...
 5.2.41: Prove that if A is a diagonalizable matrix, then the rank of A is t...
Solutions for Chapter 5.2: Diagonalization
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 5.2: Diagonalization
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. Since 41 problems in chapter 5.2: Diagonalization have been answered, more than 15824 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.2: Diagonalization includes 41 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Column space C (A) =
space of all combinations of the columns of A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.