- 9.2.1: Show that the following pairs of matrices are not similar. a. A = 2...
- 9.2.2: Show that the following pairs of matrices are not similar. a. A = 1...
- 9.2.3: Define A = PDP1 for the following matrices D and P. Determine A3. a...
- 9.2.4: Determine A4 for the matrices in Exercise 3.
- 9.2.5: For each of the following matrices determine if it diagonalizable a...
- 9.2.6: For each of the following matrices determine if it diagonalizable a...
- 9.2.7: (i) Determine if the following matrices are positive definite, and ...
- 9.2.8: (i) Determine if the following matrices are positive definite, and ...
- 9.2.9: Show that each of the following matrices is nonsingular but not dia...
- 9.2.10: Show that the following matrices are singular but are diagonalizabl...
- 9.2.11: In Exercise 31 of Section 6.6, a symmetric matrix A = 1.59 1.69 2.1...
- 9.2.12: Suppose that A and B are nonsingular n n matrices. Prove the AB is ...
- 9.2.13: Show that if A is similar to B and B is similar to C, then A is sim...
- 9.2.14: Show that if A is similar to B, then a. det(A) = det(B). b. The cha...
- 9.2.15: Show that the matrix given in Example 3 of Section 9.1, A = 2 00 1 ...
- 9.2.16: Prove Theorem 9.10.
- 9.2.17: Show that there is no diagonal matrix similar to the matrix given i...
- 9.2.18: Prove that if Q is nonsingular matrix with Qt = Q1, then Q is ortho...
- 9.2.19: Prove Theorem 9.13.
Solutions for Chapter 9.2: Orthogonal Matrices and Similarity Transformations
Full solutions for Numerical Analysis | 9th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
A sequence of steps intended to approach the desired solution.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
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).
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Outer product uv T
= column times row = rank one matrix.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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.
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.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.