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Solutions for Chapter 1.9: Leontief Input-Output Models

Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition

ISBN: 9780470432051

Solutions for Chapter 1.9: Leontief Input-Output Models

Chapter 1.9: Leontief Input-Output Models includes 16 full step-by-step solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. Since 16 problems in chapter 1.9: Leontief Input-Output Models have been answered, more than 14229 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Cholesky factorization

    A = CTC = (L.J]))(L.J]))T for positive definite A.

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • 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. Block-diagonal: zero outside square blocks Du.

  • Factorization

    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.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • 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.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • Norm

    IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

  • 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.

  • Partial pivoting.

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

  • Projection matrix P onto subspace S.

    Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B II·

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

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