 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
ISBN: 9780136009290
Solutions for Chapter 1.6: Partitioned Matrices
Get Full SolutionsSince 22 problems in chapter 1.6: Partitioned Matrices have been answered, more than 7564 students have viewed full stepbystep solutions from this chapter. Chapter 1.6: Partitioned Matrices includes 22 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290.

Big formula for n by n determinants.
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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Independent vectors VI, .. " vk.
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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pascal 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).

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
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.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.