- 4.4.1: show that v is an eigenvector of A and find 4. A = [ ! = l v = [ ] ...
- 4.4.2: Compute the determinants in Exercises 1-6 using cofactor -4 1 3 cos...
- 4.4.3: compute (a) the characteristic polynomial of A, (b) the eigenvalues...
- 4.4.4: show that A and B are not similar matrices.
- 4.4.5: a matrix A is given along with an iterate x5, produced as in Exampl...
- 4.4.6: Which of the stochastic matrices in Exercises 1-6 are regular?
Solutions for Chapter 4: Eigenvalues and Eigenvectors
Full solutions for Linear Algebra: A Modern Introduction | 1st Edition
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I 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).
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.
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.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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 = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Every v in V is orthogonal to every w in W.
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).
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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).
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.
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.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).