Solutions for Chapter 2.3: Solving Linear Equations

(Particular x p) + (x n in nullspace).

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

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

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

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

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].

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.

Blocks aij B, eigenvalues Ap(A)Aq(B).

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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 .

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Column and row spaces = lines cu and cv.

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

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}.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

Constant down each diagonal = time-invariant (shift-invariant) filter.

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.