 7.3.1: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.2: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.3: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.4: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.5: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.6: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.7: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.8: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.9: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.10: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.11: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.12: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.13: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.14: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.15: For 115, determine whether the given matrix A is diagonalizable. Wh...
 7.3.16: For 1617, use some form of technology to determine a complete set o...
 7.3.17: For 1617, use some form of technology to determine a complete set o...
 7.3.18: For 1822, use the ideas introduced in this section to solve the giv...
 7.3.19: For 1822, use the ideas introduced in this section to solve the giv...
 7.3.20: For 1822, use the ideas introduced in this section to solve the giv...
 7.3.21: For 1822, use the ideas introduced in this section to solve the giv...
 7.3.22: For 1822, use the ideas introduced in this section to solve the giv...
 7.3.23: For 2324, first write the given system of differential equations in...
 7.3.24: For 2324, first write the given system of differential equations in...
 7.3.25: Let A be a nondefective matrix. Then S1AS = D, where D is a diagona...
 7.3.26: If D = diag(1,2,...,n), show that for every positive integer k, Dk ...
 7.3.27: Use the results of the preceding two problems to determine A3 and A...
 7.3.28: We call a matrix B a square root of A if B2 = A. (a) Show that if D...
 7.3.29: Prove the following properties for similar matrices: (a) A matrix A...
 7.3.30: If A is similar to B, prove that AT is similar to BT .
 7.3.31: In Theorem 7.3.3, we proved that similar matrices have the same eig...
 7.3.32: Let A be a nondefective matrix and let S be a matrix such that S1AS...
 7.3.33: Let A be a nondefective matrix and let S be a matrix such that S1AS...
 7.3.34: If A = 2 4 1 1 , determine S such that S1AS = diag(3, 2), and use t...
 7.3.35: 3537 deal with the generalization of the diagonalization problem to...
 7.3.36: 3537 deal with the generalization of the diagonalization problem to...
 7.3.37: 3537 deal with the generalization of the diagonalization problem to...
 7.3.38: In this problem, we establish that similar matrices describe the sa...
Solutions for Chapter 7.3: Diagonalization
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 7.3: Diagonalization
Get Full SolutionsDifferential Equations was written by and is associated to the ISBN: 9780321964670. Since 38 problems in chapter 7.3: Diagonalization have been answered, more than 19145 students have viewed full stepbystep solutions from this chapter. Chapter 7.3: Diagonalization includes 38 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations, edition: 4.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.