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# Solutions for Chapter 7.1: Orthogonal Matrices

## Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version | 11th Edition

ISBN: 9781118474228

Solutions for Chapter 7.1: Orthogonal Matrices

Solutions for Chapter 7.1
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##### ISBN: 9781118474228

Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. Since 32 problems in chapter 7.1: Orthogonal Matrices have been answered, more than 16676 students have viewed full step-by-step solutions from this chapter. Chapter 7.1: Orthogonal Matrices includes 32 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11.

Key Math Terms and definitions covered in this textbook
• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• 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

• Eigenvalue A and eigenvector x.

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

• 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.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• 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.

• 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).

• Lucas numbers

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

• 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.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• Plane (or hyperplane) in Rn.

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

• 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.

• 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 ].

• Solvable system Ax = b.

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

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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