Solutions for Chapter 7.5: Rationalizing Denominators and Numerators of Radical Expressions

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

B j has b replacing column j of A; x j = det B j I det A

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Independent columns, N(A) = {O}, no free variables.

Constant along each antidiagonal; hij depends on i + j.

Diagonal entries = 1, off-diagonal entries = 0.

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 .

The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

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

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 .

A directed graph that has constants Cl, ... , Cm associated with the edges.

Any solution to Ax = b; often x p has free variables = o.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

Appears in block elimination on [~ g ].

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

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.

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.