Solutions for Chapter 2.2: MATRICES AND LINEAR TRANSFORMATIONS

peA) = det(A - AI) has peA) = zero matrix.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

cond(A) = c(A) = IIAIlIIA-III = 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.

B j has b replacing column j of A; x j = det B j I det A

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

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

Triangular matrix with one extra nonzero adjacent diagonal.

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.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Every v in V is orthogonal to every w in W.

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 •

= column times row = rank one matrix.

Any solution to Ax = b; often x p has free variables = o.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Column and row spaces = lines cu and cv.

One free variable is Si = 1, other free variables = o.

Orthonormal columns (complex analog of Q).

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