Fundamentals of Differential Equations and Boundary Value Problems 6th Edition - Solutions by Chapter

space of all combinations of the columns of A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

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

Nullspace of AT = "left nullspace" of A because y T A = OT.

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).

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

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Orthogonal Q times positive (semi)definite H.

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Column and row spaces = lines cu and cv.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

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

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

T- 1 has rank 1 above and below diagonal.

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.