Solutions for Chapter 6.4: Discrete Mathematics with Applications 4th Edition

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

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.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

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

Square root of x T x (Pythagoras in n dimensions).

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

Appears in block elimination on [~ g ].

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Real symmetric A has real A'S and orthonormal q's.

Any vector space inside V, including V and Z = {zero vector only}.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

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.