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# Solutions for Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients

## Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition

ISBN: 9781119256007

Solutions for Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients

Solutions for Chapter 7.5
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##### ISBN: 9781119256007

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients includes 25 full step-by-step solutions. Since 25 problems in chapter 7.5: Homogeneous Linear Systems with Constant Coefficients have been answered, more than 12383 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007.

Key Math Terms and definitions covered in this textbook
• Cofactor Cij.

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

• Companion matrix.

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).

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• 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.

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Graph G.

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• Inverse matrix A-I.

Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

• Reduced row echelon form R = rref(A).

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

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

T- 1 has rank 1 above and below diagonal.

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