- 2.6.6E: Repeat Exercise with and .ExerciseConsider the production model x =...
- 2.6.9E: Solve the Leontief production equation for an economy with three se...
- 2.6.7E: Let C and d be as in Exercise 5a. Determine the production level ne...
- 2.6.8E: Let C be an n × n consumption matrix whose column sums are less tha...
- 2.6.1E: Exercises 1-4 refer to an economy that is divided into three sector...
- 2.6.2E: Exercises refer to an economy that is divided into three sectors-ma...
- 2.6.3E: Exercises refer to an economy that is divided into three sectors-ma...
- 2.6.4E: Exercises refer to an economy that is divided into three sectors-ma...
- 2.6.5E: a. Compute the transfer matrix of the network in the figure below. ...
- 2.6.10E: The consumption matrix C for the U.S economy in 1972 has the proper...
- 2.6.11E: The Leontief production equation, x = Cx + d, is usually accompanie...
- 2.6.12E: Let C be a consumption matrix such that Find a difference equation ...
- 2.6.13E: [M] The consumption matrix C below is based on input–output data fo...
- 2.6.14E: [M] The demand vector in Exercise 13 is reasonable for 1958 data, b...
- 2.6.15E: [M] Use equation (6) to solve the problem in Exercise 13. Set How m...
Solutions for Chapter 2.6: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications | 5th 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.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
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
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Outer product uv T
= column times row = rank one matrix.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.