 7.3.1: In Exercises 16, determine whether the matrix is symmetric
 7.3.2: In Exercises 16, determine whether the matrix is symmetric
 7.3.3: In Exercises 16, determine whether the matrix is symmetric
 7.3.4: In Exercises 16, determine whether the matrix is symmetric
 7.3.5: In Exercises 16, determine whether the matrix is symmetric
 7.3.6: In Exercises 16, determine whether the matrix is symmetric
 7.3.7: In Exercises 714, find the eigenvalues of the symmetric matrix. For...
 7.3.8: In Exercises 714, find the eigenvalues of the symmetric matrix. For...
 7.3.9: In Exercises 714, find the eigenvalues of the symmetric matrix. For...
 7.3.10: In Exercises 714, find the eigenvalues of the symmetric matrix. For...
 7.3.11: In Exercises 714, find the eigenvalues of the symmetric matrix. For...
 7.3.12: In Exercises 714, find the eigenvalues of the symmetric matrix. For...
 7.3.13: In Exercises 714, find the eigenvalues of the symmetric matrix. For...
 7.3.14: In Exercises 714, find the eigenvalues of the symmetric matrix. For...
 7.3.15: In Exercises 1522, determine whether the matrix is orthogonal.
 7.3.16: In Exercises 1522, determine whether the matrix is orthogonal.
 7.3.17: In Exercises 1522, determine whether the matrix is orthogonal.
 7.3.18: In Exercises 1522, determine whether the matrix is orthogonal.
 7.3.19: In Exercises 1522, determine whether the matrix is orthogonal.
 7.3.20: In Exercises 1522, determine whether the matrix is orthogonal.
 7.3.21: In Exercises 1522, determine whether the matrix is orthogonal.
 7.3.22: In Exercises 1522, determine whether the matrix is orthogonal.
 7.3.23: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.24: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.25: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.26: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.27: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.28: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.29: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.30: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.31: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.32: In Exercises 2332, find an orthogonal matrix such that diagonalizes...
 7.3.33: (a) Let be an matrix. Then is symmetric if and only if is orthogona...
 7.3.34: (a) A square matrix is orthogonal if it is invertiblethat is, if (b...
 7.3.35: Prove that if is an matrix, then and are symmetric.
 7.3.36: Find and for the matrix below.
 7.3.37: Prove that if is an orthogonal matrix, then
 7.3.38: Prove that if and are orthogonal matrices, then and are orthogonal.
 7.3.39: Show that the matrix below is orthogonal for any value of
 7.3.40: Prove that if a symmetric matrix has only one eigenvalue then
 7.3.41: Prove that if A is an orthogonal matrix, then so are AT and A_1.
Solutions for Chapter 7.3: Symmetric Matrices and Orthogonal Diagonalization
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 7.3: Symmetric Matrices and Orthogonal Diagonalization
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. Since 41 problems in chapter 7.3: Symmetric Matrices and Orthogonal Diagonalization have been answered, more than 19579 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.3: Symmetric Matrices and Orthogonal Diagonalization includes 41 full stepbystep solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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).

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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}.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.