- 6.8.1: . Find the eigenvalues of each of the following matrices and verify...
- 6.8.2: Find the eigenvalues of each of the following matrices and verify t...
- 6.8.3: Find the output vector x in the open version of the Leontief inputo...
- 6.8.4: Consider the closed version of the Leontief input output model with...
- 6.8.5: Prove: If Am = O for some positive integer m, then I A is nonsingular.
- 6.8.6: Let A = 011 0 1 1 0 1 1 (a) Compute (I A) 1. (b) Compute A2 and A3....
- 6.8.7: Which of the matrices that follow are reducible? For each reducible...
- 6.8.8: Let A be a nonnegative irreducible 3 3 matrix whose eigenvalues sat...
- 6.8.9: Let A = B O O C where B and C are square matrices. (a) If is an eig...
- 6.8.10: Prove that a 2 2 matrix A is reducible if and only if a12a21 = 0.
- 6.8.11: Prove the Frobenius theorem in the case where A is a 2 2 matrix.
- 6.8.12: We can show that, for an n n stochastic matrix, 1 = 1 is an eigenva...
- 6.8.13: Let A be an n n positive stochastic matrix with dominant eigenvalue...
- 6.8.14: Would the results of parts (c) and (d) in Exercise 13 be valid if t...
- 6.8.15: A management student received fellowship offers from four universit...
Solutions for Chapter 6.8: Nonnegative Matrices
Full solutions for Linear Algebra with Applications | 9th Edition
Tv = Av + Vo = linear transformation plus shift.
peA) = det(A - AI) has peA) = zero matrix.
Remove row i and column j; multiply the determinant by (-I)i + j •
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.
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
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
A sequence of steps intended to approach the desired solution.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
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).
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).