Solutions for Chapter 2.3: Discrete Mathematics and Its Applications 7th Edition

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

(Particular x p) + (x n in nullspace).

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

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

Invert A by row operations on [A I] to reach [I A-I].

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 .

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Blocks aij B, eigenvalues Ap(A)Aq(B).

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

Orthogonal Q times positive (semi)definite H.

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

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

T- 1 has rank 1 above and below diagonal.

Orthonormal columns (complex analog of Q).

The rows (or the columns) of A generate a box with volume I det(A) I.

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).