 7.1.1E: Determine which of the matrices in Exercises 1–6 are symmetric.
 7.1.2E: Determine which of the matrices in Exercises 1–6 are symmetric.
 7.1.3E: Determine which of the matrices in Exercises 1–6 are symmetric.
 7.1.4E: Determine which of the matrices in Exercises 1–6 are symmetric.
 7.1.5E: Determine which of the matrices in Exercises 1–6 are symmetric.
 7.1.6E: Determine which of the matrices in Exercises 1–6 are symmetric.
 7.1.7E: Determine which of the matrices in Exercises 7–12 are orthogonal. I...
 7.1.8E: Determine which of the matrices in Exercises 7–12 are orthogonal. I...
 7.1.9E: Determine which of the matrices in Exercises 7–12 are orthogonal. I...
 7.1.10E: Determine which of the matrices in Exercises 7–12 are orthogonal. I...
 7.1.11E: Determine which of the matrices in Exercises 7–12 are orthogonal. I...
 7.1.12E: Determine which of the matrices in Exercises 7–12 are orthogonal. I...
 7.1.13E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.14E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.15E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.16E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.17E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.18E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.19E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.20E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.21E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.22E: Orthogonally diagonalize the matrices in Exercises 13–22, giving an...
 7.1.23E: Verify that 2 is an eigenvalue of A and v is an eigenvector. Then o...
 7.1.24E: Verify that v1 and v2 are eigenvectors of A. Then orthogonally diag...
 7.1.25E: In Exercises 25 and 26, mark each statement True or False. Justify ...
 7.1.26E: In Exercises 25 and 26, mark each statement True or False. Justify ...
 7.1.27E: Suppose A is a symmetric n × n matrix and B is any n × m matrix. Sh...
 7.1.28E: Show that if A is an n × n symmetric matrix, then
 7.1.29E: Suppose A is invertible and orthogonally diagonalizable. Explain wh...
 7.1.30E: Suppose A and B are both orthogonally diagonalizable and AB = BA. E...
 7.1.31E: Let A = PDP–1, where P is orthogonal and D is diagonal, and let be ...
 7.1.32E: Suppose A = PRP–1, where P is orthogonal and R is upper triangular....
 7.1.33E: Construct a spectral decomposition of A from Example 2.Reference:If...
 7.1.34E: Construct a spectral decomposition of A from Example 3.Reference:
 7.1.35E: Let u be a unit vector in Rn, and let B = uuT .a. Given any x in Rn...
 7.1.36E: Let B be an n × n symmetric matrix such that B2 = B. Any such matri...
 7.1.37E: [M] Orthogonally diagonalize the matrices in Exercises 37–40. To pr...
 7.1.38E: [M] Orthogonally diagonalize the matrices in Exercises 37–40. To pr...
 7.1.39E: [M] Orthogonally diagonalize the matrices in Exercises 37–40. To pr...
 7.1.40E: [M] Orthogonally diagonalize the matrices in Exercises 37–40. To pr...
Solutions for Chapter 7.1: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 7.1
Get Full SolutionsLinear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This expansive textbook survival guide covers the following chapters and their solutions. Since 40 problems in chapter 7.1 have been answered, more than 32646 students have viewed full stepbystep solutions from this chapter. Chapter 7.1 includes 40 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

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].

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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).

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.