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Solutions for Chapter 8.1: Stable Age Distribution in a Population; Markov Processes

Elementary Linear Algebra with Applications | 9th Edition | ISBN: 9780132296540 | Authors: Bernard Kolman David Hill

Full solutions for Elementary Linear Algebra with Applications | 9th Edition

ISBN: 9780132296540

Elementary Linear Algebra with Applications | 9th Edition | ISBN: 9780132296540 | Authors: Bernard Kolman David Hill

Solutions for Chapter 8.1: Stable Age Distribution in a Population; Markov Processes

This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Chapter 8.1: Stable Age Distribution in a Population; Markov Processes includes 13 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 13 problems in chapter 8.1: Stable Age Distribution in a Population; Markov Processes have been answered, more than 13627 students have viewed full step-by-step solutions from this chapter. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • 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.

  • 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

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • Diagonalization

    A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • Identity matrix I (or In).

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

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • Nilpotent matrix N.

    Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Random matrix rand(n) or randn(n).

    MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Schwarz inequality

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

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

  • Transpose matrix AT.

    Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

  • 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·

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).