 8.1.1: For each of the matrices in Exercises 1 through 6, find an orthonor...
 8.1.2: For each of the matrices in Exercises 1 through 6, find an orthonor...
 8.1.3: For each of the matrices in Exercises 1 through 6, find an orthonor...
 8.1.4: For each of the matrices in Exercises 1 through 6, find an orthonor...
 8.1.5: For each of the matrices in Exercises 1 through 6, find an orthonor...
 8.1.6: For each of the matrices in Exercises 1 through 6, find an orthonor...
 8.1.7: For each of the matrices A in Exercises 7 through 11, find an ortho...
 8.1.8: For each of the matrices A in Exercises 7 through 11, find an ortho...
 8.1.9: For each of the matrices A in Exercises 7 through 11, find an ortho...
 8.1.10: For each of the matrices A in Exercises 7 through 11, find an ortho...
 8.1.11: For each of the matrices A in Exercises 7 through 11, find an ortho...
 8.1.12: Let L from spanned bya. Find an orthonormal eigenbasis 53 for L. b....
 8.1.13: Consider a symmetric 3 x 3 matrix A with A2 = I3. Is the linear tra...
 8.1.14: In Example 3 of this section, we diagonalized the matrix1 1 f 1 1 1...
 8.1.15: If A is invertible and orthogonally diagonalizable, is A 1 orthogon...
 8.1.16: a. Find the eigenvalues of the matrix4 ="1 1 1 1" 1 1 1 1 1 1 1 1 1...
 8.1.17: Use the approach of Exercise 16 to find the determinant of the n x ...
 8.1.18: Consider unit vectors v\,..., vn in R" such that the angle between ...
 8.1.19: Consider a linear transformation L from Rm to Rn. Show that there e...
 8.1.20: Consider a linear transformation T from Rm to Rn, where m <n. Show ...
 8.1.21: Consider a symmetric 3x3 matrix A with eigenvalues 1, 2, and 3. How...
 8.1.22: Consider the matrixA =where k is a constant. a. Find a value of k s...
 8.1.23: If an n x n matrix A is both symmetric and orthogonal, what can you...
 8.1.24: Consider the matrixFind an orthonormal eigenbasis for A
 8.1.25: Consider the matrixFind an orthogonal 5 x 5 matrix S such that S di...
 8.1.26: Let Jn be the n x n matrix with all ones on the other diagonal and ...
 8.1.27: Diagonalize the n x n matrix(All ones along both diagonals,;and zer...
 8.1.28: Diagonalize the 13 x 13 matrix (All ones in the last row and the la...
 8.1.29: Consider a symmetric matrix A. If the vector v is in the image of A...
 8.1.30: Consider an orthogonal matrix R whose first column is v. Form the s...
 8.1.31: True orf alse! If A is a symmetric matrix, then rank (A) = rank(A2).
 8.1.32: Consider the n x n matrix with all ones on the main diagonal and al...
 8.1.33: For which angle(s) 0 can you find three distinct unit vectors in R2...
 8.1.34: For which angle(s) 0 can you find four distinct unit vectors in R3 ...
 8.1.35: Consider n + 1 distinct unit vectors in R" such that the angle betw...
 8.1.36: Consider a symmetric n x n matrix A with A2 = A. Is the linear tran...
 8.1.37: If A is any symmetric 2x2 matrix with eigenvalues 2 and 3, and u is...
 8.1.38: If A is any symmetric 2 x 2 matrix with eigenvalues 2 and 3, and u ...
 8.1.39: If A is any symmetric 3 x 3 matrix with eigenvalues 2, 3, and 4, an...
 8.1.40: If A is any symmetric 3x3 matrix with eigenvalues 2, 3, and 4, and ...
 8.1.41: Show that for every symmetric n x n matrix A there exists a symmetr...
 8.1.42: Find a symmetric 2 x 2 matrix B such that
 8.1.43: For A =find a nonzero vector v in ] such that Ai) is orthogonal to v
 8.1.44: Consider an invertible symmetric nxn matrix A. When does there exis...
 8.1.45: We say that an nxn matrix A is triangulizable if A is similar to an...
 8.1.46: a. Consider a complex upper triangular nxn matrix U with zeros on t...
 8.1.47: Let us first introduce two notations. For a complex n x n matrix A,...
 8.1.48: Let U > 0 be a real upper triangular n x n matrix with zeros on the...
 8.1.49: Let R be a complex upper triangular nxn matrix with  rn  < 1 for ...
 8.1.50: a. Let A be a complex nxn matrix such that A. < j for all eigenva...
Solutions for Chapter 8.1: Symmetric Matrices
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 8.1: Symmetric Matrices
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 50 problems in chapter 8.1: Symmetric Matrices have been answered, more than 14228 students have viewed full stepbystep solutions from this chapter. Chapter 8.1: Symmetric Matrices includes 50 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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 nl  An).

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.