 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
ISBN: 9780321962218
Solutions for Chapter 6.8: Nonnegative Matrices
Get Full SolutionsChapter 6.8: Nonnegative Matrices includes 15 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218. Since 15 problems in chapter 6.8: Nonnegative Matrices have been answered, more than 10827 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

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.

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.

Iterative method.
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.

lAII = 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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Partial pivoting.
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 = MI AM has the same eigenvalues as A.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).