Solutions for Chapter 15: Advanced Engineering Mathematics 9th Edition

The n roots are the eigenvalues of A.

A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

Ax = AX with x#-O so det(A - AI) = o.

Columns without pivots; these are combinations of earlier columns.

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

A symmetric matrix with eigenvalues of both signs (+ and - ).

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

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

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

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

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Each equation gives a plane in Rn; the planes intersect at x.

Appears in block elimination on [~ g ].

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

The right side b is in the column space of A.

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

Signs in A = signs in D.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.