- 1.6.1: Let A be a nonsingular n n matrix. Perform the following multiplica...
- 1.6.2: Let B = ATA. Show that bi j = 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) 1 1 1 1 2 ...
- 1.6.6: Given X = 2 1 5 4 2 3 Y = 1 2 4 2 3 1 (a) Compute the outer product...
- 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 wh...
- 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 kk. Determine A2 and...
- 1.6.14: Let I denote the nn identity matrix. Find a block form for the inve...
- 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 n n matrices with the property that Bx = Cx for all ...
- 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 | 8th Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Upper triangular systems are solved in reverse order Xn to Xl.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
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).
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.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear transformation T.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Outer product uv T
= column times row = rank one matrix.
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.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.