 9.9.1: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.25: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.41: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.2: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.26: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.42: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.3: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.27: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.43: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.4: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.28: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.44: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.5: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.29: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.45: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.6: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.30: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.46: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.7: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.31: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.47: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.8: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.32: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.48: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.9: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.33: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.49: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.10: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.34: In each of 1 through 10, produce a matrix P that diagonalizes the g...
 9.9.50: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.11: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.35: Let A have eigenvalues 1, , n , and suppose that P diagonalizes A. ...
 9.9.51: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.12: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.36: In each of 12 through 15, use the idea of to compute the indicated ...
 9.9.52: In each of 1 through 12, find the eigenvalues and associated eigenv...
 9.9.13: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.37: In each of 12 through 15, use the idea of to compute the indicated ...
 9.9.53: In each of 13 through 21, determine whether the matrix is unitary, ...
 9.9.14: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.38: In each of 12 through 15, use the idea of to compute the indicated ...
 9.9.54: In each of 13 through 21, determine whether the matrix is unitary, ...
 9.9.15: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.39: In each of 12 through 15, use the idea of to compute the indicated ...
 9.9.55: In each of 13 through 21, determine whether the matrix is unitary, ...
 9.9.16: In each of 1 through 16, find the eigenvalues of the matrix. For ea...
 9.9.40: Suppose A2 is diagonalizable. Prove that A is diagonalizable
 9.9.56: In each of 13 through 21, determine whether the matrix is unitary, ...
 9.9.17: In each of 17 through 22, find the eigenvalues and associated eigen...
 9.9.57: In each of 13 through 21, determine whether the matrix is unitary, ...
 9.9.18: In each of 17 through 22, find the eigenvalues and associated eigen...
 9.9.58: In each of 13 through 21, determine whether the matrix is unitary, ...
 9.9.19: In each of 17 through 22, find the eigenvalues and associated eigen...
 9.9.59: In each of 13 through 21, determine whether the matrix is unitary, ...
 9.9.20: In each of 17 through 22, find the eigenvalues and associated eigen...
 9.9.60: In each of 13 through 21, determine whether the matrix is unitary, ...
 9.9.21: In each of 17 through 22, find the eigenvalues and associated eigen...
 9.9.61: In each of 13 through 21, determine whether the matrix is unitary, ...
 9.9.22: In each of 17 through 22, find the eigenvalues and associated eigen...
 9.9.62: In each of 22 through 28, determine a matrix A so that the quadrati...
 9.9.23: Suppose is an eigenvalue of A with eigenvector E. Let k be a positi...
 9.9.63: In each of 22 through 28, determine a matrix A so that the quadrati...
 9.9.24: Let A be an n n matrix of numbers. Show that the constant term in t...
 9.9.64: In each of 22 through 28, determine a matrix A so that the quadrati...
 9.9.65: In each of 22 through 28, determine a matrix A so that the quadrati...
 9.9.66: In each of 22 through 28, determine a matrix A so that the quadrati...
 9.9.67: In each of 22 through 28, determine a matrix A so that the quadrati...
 9.9.68: In each of 22 through 28, determine a matrix A so that the quadrati...
 9.9.69: Suppose A is hermitian. Show that (AAt ) = AA.
 9.9.70: Prove that the main diagonal elements of a hermitian matrix are real.
 9.9.71: Prove that each main diagonal element of a skewhermitian matrix is ...
 9.9.72: Prove that the product of two unitary matrices is unitary
Solutions for Chapter 9: Eigenvalues, Diagonalization, and Special Matrices
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 9: Eigenvalues, Diagonalization, and Special Matrices
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. This expansive textbook survival guide covers the following chapters and their solutions. Since 72 problems in chapter 9: Eigenvalues, Diagonalization, and Special Matrices have been answered, more than 16496 students have viewed full stepbystep solutions from this chapter. Chapter 9: Eigenvalues, Diagonalization, and Special Matrices includes 72 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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).

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.