- 2.1: Generate random 55 matrices with integer entries by setting A = rou...
- 2.2: Are n n magic squares nonsingular? Use the MATLAB command det(magic...
- 2.3: Set A = round(10 rand(6)). In each of the following, use MATLAB to ...
- 2.4: We can generate a random 6 6 matrix A whose entries consist entirel...
- 2.5: If a matrix is sensitive to roundoff errors, the computed value of ...
- 2.6: Use MATLAB to construct a matrix A by setting A = vander(1 : 6); A ...
- 2.7: A triangular matrix is nonsingular if and only if its diagonal entr...
- 2.8: If x is a nonzero vector in Rn and Ax = 0, then det(A) = 0.
- 2.9: If A and B are row equivalent matrices, then their determinants are...
- 2.10: If A _= O, but Ak = O (where O denotes the zero matrix) for some po...
Solutions for Chapter 2: Determinants
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.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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.
Determinant IAI = det(A).
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
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
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).
Normal equation AT Ax = ATb.
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Outer product uv T
= column times row = rank one matrix.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Solvable system Ax = b.
The right side b is in the column space of A.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.