- 1.6.1: Let A be a nonsingular n n matrix. Perform the following multiplica...
- 1.6.2: Let B = AT A. Show that bij = aT i aj.
- 1.6.3: Let A = 1 1 2 1 and B = 2 1 1 3 (a) Calculate Ab1 and Ab2. (b) Calc...
- 1.6.4: Let I = 1 0 0 1 , E = 0 1 1 0 , O = 0 0 0 0 C = 1 0 1 1 , D = 2 0 0...
- 1.6.5: Perform each of the following block multiplications: (a) 111 1 212 ...
- 1.6.6: Given X = 215 423 , Y = 124 231 (a) Compute the outer product expan...
- 1.6.7: Let A = A11 A12 A21 A22 and AT = AT 11 AT 21 AT 12 AT 22 Is it poss...
- 1.6.8: Let A be an m n matrix, X an n r matrix, and B an m r matrix. Show ...
- 1.6.9: Let A be an n n matrix and let D be an n n diagonal matrix. (a) Sho...
- 1.6.10: Let U be an m m matrix, let V be an n n matrix, and let = 1 O where...
- 1.6.11: Let A = A11 A12 O A22 where all four blocks are n n matrices. (a) I...
- 1.6.12: Let A and B be n n matrices and let M be a block matrix of the form...
- 1.6.13: Let A = O I B O where all four submatrices are k k. Determine A2 an...
- 1.6.14: Let I denote the n n identity matrix. Find a block form for the inv...
- 1.6.15: Let O be the k k matrix whose entries are all 0, I be the k k ident...
- 1.6.16: Let A and B be n n matrices and define 2n 2n matrices S and M by S ...
- 1.6.17: Let A = A11 A12 A21 A22 where A11 is a k k nonsingular matrix. Show...
- 1.6.18: Let A, B, L, M, S, and T be n n matrices with A, B, and M nonsingul...
- 1.6.19: Let A be an n n matrix and x Rn. (a) A scalar c can also be conside...
- 1.6.20: If A is an nn matrix with the property that Ax = 0 for all x Rn, sh...
- 1.6.21: Let B and C be nn matrices with the property that Bx = Cx for all x...
- 1.6.22: Consider a system of the form A a cT x xn+1 = b bn+1 where A is a n...
Solutions for Chapter 1.6: Partitioned Matrices
Full solutions for Linear Algebra with Applications | 9th Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Elimination matrix = Elementary matrix Eij.
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.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
= Xl (column 1) + ... + xn(column n) = combination of columns.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Vandermonde matrix V.
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.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.