Solutions for Chapter 25: Miscellaneous questions

Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Tv = Av + Vo = linear transformation plus shift.

Parentheses can be removed to leave ABC.

If diagonalizable, they share n eigenvectors.

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

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 .

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.

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

The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

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).

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.

Column and row spaces = lines cu and cv.

= number of pivots = dimension of column space = dimension of row space.

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Appears in block elimination on [~ g ].

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

Signs in A = signs in D.