 11.5.1: In Exercises 16, find A for the given A.A =1 + i 3i2 i 1 + 4i2
 11.5.2: In Exercises 16, find A for the given A.A = 7i 3 2i1 + 5i 83
 11.5.3: In Exercises 16, find A for the given A.A = 3 + i 5i 1 i1 4i 8 6 + ...
 11.5.4: In Exercises 16, find A for the given A.A = 5 i 2 + 7i4i 5i 315 i 6...
 11.5.5: In Exercises 16, find A for the given A.A =1 2i 3 4i2i 5 6i 1 + i3 ...
 11.5.6: In Exercises 16, find A for the given A.A =4 1 2i 12 1 + 3i11i 6 5i...
 11.5.7: In Exercises 712, determine if the given matrix is Hermitian.A =1 +...
 11.5.8: In Exercises 712, determine if the given matrix is Hermitian.A = 4 ...
 11.5.9: In Exercises 712, determine if the given matrix is Hermitian.A = 3 ...
 11.5.10: In Exercises 712, determine if the given matrix is Hermitian.A = 5 ...
 11.5.11: In Exercises 712, determine if the given matrix is Hermitian.A =1 2...
 11.5.12: In Exercises 712, determine if the given matrix is Hermitian.A =0 1...
 11.5.13: In Exercises 1318, determine if the given matrix is normal.A = 1 2 ...
 11.5.14: In Exercises 1318, determine if the given matrix is normal.A = 3 3 ...
 11.5.15: In Exercises 1318, determine if the given matrix is normal.A =i ii i1
 11.5.16: In Exercises 1318, determine if the given matrix is normal.A =2i ii 3i
 11.5.17: In Exercises 1318, determine if the given matrix is normal.A = 1 i ...
 11.5.18: In Exercises 1318, determine if the given matrix is normal.A = 2 3 ...
 11.5.19: FIND AN EXAMPLE For Exercises 1922, find an example thatmeets the g...
 11.5.20: FIND AN EXAMPLE For Exercises 1922, find an example thatmeets the g...
 11.5.21: FIND AN EXAMPLE For Exercises 1922, find an example thatmeets the g...
 11.5.22: FIND AN EXAMPLE For Exercises 1922, find an example thatmeets the g...
 11.5.23: TRUE OR FALSE For Exercises 2330, determine if the statementis true...
 11.5.24: TRUE OR FALSE For Exercises 2330, determine if the statementis true...
 11.5.25: TRUE OR FALSE For Exercises 2330, determine if the statementis true...
 11.5.26: TRUE OR FALSE For Exercises 2330, determine if the statementis true...
 11.5.27: TRUE OR FALSE For Exercises 2330, determine if the statementis true...
 11.5.28: TRUE OR FALSE For Exercises 2330, determine if the statementis true...
 11.5.29: TRUE OR FALSE For Exercises 2330, determine if the statementis true...
 11.5.30: TRUE OR FALSE For Exercises 2330, determine if the statementis true...
 11.5.31: Prove that if A has real entries, then A = AT .
 11.5.32: Prove that if A has complex entries, then (A)T = (AT ). Thisshows t...
 11.5.33: In Exercises 3336, suppose that A and B are matrices with complexen...
 11.5.34: In Exercises 3336, suppose that A and B are matrices with complexen...
 11.5.35: In Exercises 3336, suppose that A and B are matrices with complexen...
 11.5.36: In Exercises 3336, suppose that A and B are matrices with complexen...
 11.5.37: Show that a square matrix A is unitary if and only if thecolumns of...
 11.5.38: Prove that any Hermitian matrix must have real diagonal entries
 11.5.39: Show that if A is upper (or lower) triangular and normal, thenA mus...
 11.5.40: Suppose that A = PDP 1, where D is diagonal and P isunitary. Show t...
Solutions for Chapter 11.5: Hermitian Matrices
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 11.5: Hermitian Matrices
Get Full SolutionsChapter 11.5: Hermitian Matrices includes 40 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 40 problems in chapter 11.5: Hermitian Matrices have been answered, more than 16821 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.)

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

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

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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).

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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