×
Get Full Access to Math - Textbook Survival Guide
Get Full Access to Math - Textbook Survival Guide
×

# Solutions for Chapter 4.7: Homogeneous Systems

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

ISBN: 9780132296540

Solutions for Chapter 4.7: Homogeneous Systems

Solutions for Chapter 4.7
4 5 0 401 Reviews
21
3
##### ISBN: 9780132296540

Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. Since 26 problems in chapter 4.7: Homogeneous Systems have been answered, more than 12033 students have viewed full step-by-step solutions from this chapter. Chapter 4.7: Homogeneous Systems includes 26 full step-by-step solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Complex conjugate

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

• Conjugate Gradient Method.

A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• Diagonal matrix D.

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

• 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.

• Ellipse (or ellipsoid) x T Ax = 1.

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• Full column rank r = n.

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

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• Indefinite matrix.

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

• 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.

• Normal equation AT Ax = ATb.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Saddle point of I(x}, ... ,xn ).

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Solvable system Ax = b.

The right side b is in the column space of A.