Linear Algebra: A Geometric Approach 2nd Edition - Solutions by Chapter

If diagonalizable, they share n eigenvectors.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

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

Constant along each antidiagonal; hij depends on i + 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.

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.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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 .

IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

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

= column times row = rank one matrix.

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.

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Every B = M-I AM has the same eigenvalues as A.

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

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

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Orthonormal columns (complex analog of Q).