- 9.2.1: Show that the following pairs of matrices are not similar. 2 1 1 2 ...
- 9.2.2: Show that the following pairs of matrices are not similar. a. A = b...
- 9.2.3: Define A = PDP for the following matrices D and P. Determine A . 4....
- 9.2.4: Determine A 4 for the matrices in Exercise 3
- 9.2.5: For each of the following matrices, determine if it is diagonalizab...
- 9.2.6: For each of the following matrices, determine if it is diagonalizab...
- 9.2.7: For the matrices in Exercise I of Section 9.1 that have three linea...
- 9.2.8: For the matrices in Exercise 2 of Section 9.1 that have three linea...
- 9.2.9: (i) Determine if the following matrices are positive definite and, ...
- 9.2.10: (i) Determine if the following matrices are positive definite and, ...
- 9.2.11: Show that each ofthe following matrices is nonsingular but not diag...
- 9.2.12: Show that the following matrices are singular but are diagonalizabl...
- 9.2.13: Show that the matrix given in Example 3 of Section 9.1, " 2 0 0 " A...
- 9.2.14: Show that there is no diagonal matrix similar to the matrix given i...
- 9.2.15: In Exercise 22 of Section 6.6, a symmetric matrix A = 1.59 1.69 2.1...
- 9.2.16: Suppose that A and B are nonsingular n x n matrices. Prove that AS ...
- 9.2.17: Show that if A is similar to B and B is similar to C, then A is sim...
- 9.2.18: Show thatif A is similar to B, then a. det(A) = det(fi). The charac...
- 9.2.19: Prove Theorem 9.10.
- 9.2.20: Prove Theorem 9.13.
Solutions for Chapter 9.2: Orthogonal Matrices and Similarity Transformations
Full solutions for Numerical Analysis | 10th Edition
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.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
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.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
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.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Every v in V is orthogonal to every w in W.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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.