Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
Find the lengths of u = (1 + i, 1 - i, 1 + 2i) and v = (i, i, i). Also find uHv and vHu.

Introduction to Linear Algebra 4
Compute AHA and AAH. Those are both __ matrices: A=[~ : ~l

Introduction to Linear Algebra 4
Solve Az = 0 to find a vector in the nullspace of A in Problem 2. Show that z is orthogonal to the columns of AH. Show that z is not orthogonal to the columns of AT. The good row space is no longer C (AT). Now it is C (AH).

Introduction to Linear Algebra 4
Problem 3 indicates that the four fundamental subspaces are C(A) and N(A) and __ and . Their dimensions are still rand n - rand rand m - r. They are still orthogonal subspaces. The symbol H takes the place ofT

Introduction to Linear Algebra 4
(a) Prove that A H A is always a Hermitian matrix. (b) If Az = 0 then AH Az = O. If AH Az = 0, multiply by zH to prove that Az = O. The nullspaces of A and AHA are . Therefore AHA is an invertible Hermitian matrix when the nullspace of A contains only z = 0.

Introduction to Linear Algebra 4
True or false (give a reason if true or a counterexample if false): (a) If A is a real matrix then A + il is invertible. (b) If A is a Hermitian matrix then A + i 1 is invertible. (c) If U is a unitary matrix then A + if is invertible.

Introduction to Linear Algebra 4
When you mUltiply a Hermitian matrix by a real number c, is cA still Hermitian? Show that iA is skew-Hermitian when A is Hermitian. The 3 by 3 Hermitian matrices are a subspace provided the "scalars" are real numbers.

Introduction to Linear Algebra 4
Which classes of matrices does P belong to: invertible, Hermitian, unitary? P= [ i . 0 i 0] i Compute P 2, P 3, and P 100. What are the eigenvalues of P?

Introduction to Linear Algebra 4
Find the unit eigenvectors of P in Problem 8, and put them into the columns of a unitary matrix F. What property of P makes these eigenvectors orthogonal?

Introduction to Linear Algebra 4
Write down the 3 by 3 circulant matrix C = 21 + SP. It has the same eigenvectors as P in Problem 8. Find its eigenvalues.

Introduction to Linear Algebra 4
If U and V are unitary matrices, show that U-1 is unitary and also U V is unitary. Start from UHU = 1 and VHV = I.

Introduction to Linear Algebra 4
How do you know that the determinant of every Hermitian matrix is real?

Introduction to Linear Algebra 4
The matrix AHA is not only Hermitian but also positive definite, when the columns of A are independent. Proof: zH A H Az is positive if z is nonzero because __

Introduction to Linear Algebra 4
Diagonalize this Hermitian matrix to reach A = UAUH : 1 - i] 1

Introduction to Linear Algebra 4
Diagonalize this skew-Hermitian matrix to reach K = U AUH All A'S are __ K = [I ~ i -\+ i l

Introduction to Linear Algebra 4
Diagonalize this orthogonal matrix to reach Q = U AUH Now all A'S are __ 17 18 19 Q = [C?S (J - sin (J]

Introduction to Linear Algebra 4
Diagonalize this unitary matrix V to reach V = U AUH Again all A'S are __ I [ 1 V=,J3 I+i 1 - i] -I .

Introduction to Linear Algebra 4
If VI, ... ,Vn is an orthonormal basis for en, the matrix with those columns is a __ matrix. Show that any vector Z equals (VrZ)Vl + ... + (v~z)vn.

Introduction to Linear Algebra 4
The functions e-ix and eix are orthogonal on the interval 0 < x < 2n because their . d r2:n: 0 mner pro uct IS Jo = .

Introduction to Linear Algebra 4
The vectors v = (1, i, 1), w = (i, 1, 0) and Z = __ are an orthogonal basis for

Introduction to Linear Algebra 4
If A = R + iSis a Hermitian matrix, are its real and imaginary parts symmetric?

Introduction to Linear Algebra 4
The (complex) dimension of en is __ . Find a non-real basis for en

Introduction to Linear Algebra 4
Describe all 1 by 1 and 2 by 2 Hermitian matrices and unitary matrices

Introduction to Linear Algebra 4
How are the eigenvalues of A H related to the eigenvalues of the square complex matrix A?

Introduction to Linear Algebra 4
If uHu = 1 show that J - 2uuH is Hermitian and also unitary. The rank-one matrix uuH is the projection onto what line in en?

Introduction to Linear Algebra 4
If A + iB is a unitary matrix (A and B are real) show that Q = [~-!] is an orthogonal matrix.

Introduction to Linear Algebra 4
If A + iB is Hermitian (A and B are real) show that [~ -!] is symmetric.

Introduction to Linear Algebra 4
Prove that the inverse of a Hermitian matrix is also Hermitian (transpose A-I A = J).

Introduction to Linear Algebra 4
Diagonalize this matrix by constructing its eigenvalue matrix A and its eigenvector matrix S: A=[l!i l~i]=AH.

Introduction to Linear Algebra 4
A matrix with orthonormal eigenvectors has the form A = U A U- 1 = U AU H. Prove that AAH = AHA. These are exactly the normal matrices. Examples are Hermitian, skew-Hermitian, and unitary matrices. Construct a 2 by 2 normal matrix by choosing complex eigenvalues in A.