Solutions for Chapter 3.1: Discrete Mathematics and Its Applications 7th Edition

Parentheses can be removed to leave ABC.

Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

has derivative AeAt; eAt u(O) solves u' = Au.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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 .

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.

A combination other than all Ci = 0 gives L Ci Vi = O.

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.

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).

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

P = aaT laTa has rank l.

Appears in block elimination on [~ g ].

Every B = M-I AM has the same eigenvalues as A.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Columns of n by n identity matrix (written i ,j ,k in R3).

Constant down each diagonal = time-invariant (shift-invariant) filter.