8.7E: Complex Matrices Exercises

Exercises for 1

solutions

2

In each case, compute the norm of the complex vector.

1. \((1, 1 - i, -2, i)\)
2. \((1 - i, 1 + i, 1, -1)\)
3. \((2 + i, 1 - i, 2, 0, -i)\)
4. \((-2, -i, 1 + i, 1 - i, 2i)\)

2. \(\sqrt{6}\)
3. \(\sqrt{13}\)

In each case, determine whether the two vectors are orthogonal.

1. \((4, -3i, 2 + i)\), \((i, 2, 2 - 4i)\)
2. \((i, -i, 2 + i)\), \((i, i, 2 - i)\)
3. \((1, 1, i, i)\), \((1, i, -i, 1)\)
4. \((4 + 4i, 2 + i, 2i)\), \((-1 + i, 2, 3 - 2i)\)

2. Not orthogonal
3. Orthogonal

A subset \(U\) of \(\mathbb{C}^n\) is called a complex subspace of \(\mathbb{C}^n\) if it contains \(0\) and if, given \(\mathbf{v}\) and \(\mathbf{w}\) in \(U\), both \(\mathbf{v} + \mathbf{w}\) and \(z\mathbf{v}\) lie in \(U\) (\(z\) any complex number). In each case, determine whether \(U\) is a complex subspace of \(\mathbb{C}^3\).

1. \(U = \{(w, \overline{w}, 0) \mid w \mbox{ in } \mathbb{C}\}\)
2. \(U = \{(w, 2w, a) \mid w \mbox{ in } \mathbb{C}, a \mbox{ in } \mathbb{R}\}\)
3. \(U = \mathbb{R}^3\)
4. \(U = \{(v + w, v - 2w, v) \mid v, w \mbox{ in } \mathbb{C}\}\)

2. Not a subspace. For example, \(i(0, 0, 1) = (0, 0, i)\) is not in \(U\).
3. This is a subspace.

In each case, find a basis over \(\mathbb{C}\), and determine the dimension of the complex subspace \(U\) of \(\mathbb{C}^3\) (see the previous exercise).

1. \(U = \{(w, v + w, v - iw) \mid v, w \mbox{ in } \mathbb{C}\}\)
2. \(U = \{(iv + w, 0, 2v - w) \mid v, w \mbox{ in } \mathbb{C}\}\)
3. \(U = \{(u, v, w) \mid iu - 3v + (1 - i)w = 0;\ \\ u, v, w \mbox{ in } \mathbb{C}\}\)
4. \(U = \{(u, v, w) \mid 2u + (1 + i)v - iw = 0;\ \\ u, v, w \mbox{ in } \mathbb{C}\}\)

2. Basis \(\{(i, 0, 2), (1, 0, -1)\}\); dimension \(2\)
3. Basis \(\{(1, 0, -2i), (0, 1, 1 - i)\}\); dimension \(2\)

In each case, determine whether the given matrix is hermitian, unitary, or normal.

\(\left[ \begin{array}{rr} 1 & -i \\ i & i \end{array}\right]\) \(\left[ \begin{array}{rr} 2 & 3 \\ -3 & 2 \end{array}\right]\) \(\left[ \begin{array}{rr} 1 & i \\ -i & 2 \end{array}\right]\) \(\left[ \begin{array}{rr} 1 & -i \\ i & -1 \end{array}\right]\) \(\frac{1}{\sqrt{2}} \left[ \begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right]\) \(\left[ \begin{array}{cc} 1 & 1 + i \\ 1 + i & i \end{array}\right]\) \(\left[ \begin{array}{cc} 1 + i & 1 \\ -i & -1 + i \end{array}\right]\) \(\frac{1}{\sqrt{2}|z|}\left[ \begin{array}{rr} z & z \\ \overline{z} & -\overline{z} \end{array}\right]\), \(z \neq 0\)

2. Normal only
3. Hermitian (and normal), not unitary
4. None
5. Unitary (and normal); hermitian if and only if \(z\) is real

Show that a matrix \(N\) is normal if and only if \(\overline{N}N^T = N^T\overline{N}\).

Let \(A = \left[ \begin{array}{cc} z & \overline{v} \\ v & w \end{array}\right]\) where \(v\), \(w\), and \(z\) are complex numbers. Characterize in terms of \(v\), \(w\), and \(z\) when \(A\) is

hermitian unitary normal.

In each case, find a unitary matrix \(U\) such that \(U^{H}AU\) is diagonal.

1. \(A = \left[ \begin{array}{rr} 1 & i\\ -i & 1 \end{array}\right]\)
2. \(A = \left[ \begin{array}{cc} 4 & 3 - i \\ 3 + i & 1 \end{array}\right]\)
3. \(A = \left[ \begin{array}{rr} a & b\\ -b & a \end{array}\right]\); \(a\), \(b\), real
4. \(A = \left[ \begin{array}{cc} 2 & 1 + i\\ 1 - i & 3 \end{array}\right]\)
5. \(A = \left[ \begin{array}{ccc} 1 & 0 & 1 + i\\ 0 & 2 & 0 \\ 1 - i & 0 & 0 \end{array}\right]\)
6. \(A = \left[ \begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 1 + i\\ 0 & 1 - i & 2 \end{array}\right]\)

2. \(U = \frac{1}{\sqrt{14}}\left[ \begin{array}{cc} -2 & 3 - i \\ 3 + i & 2 \end{array}\right]\), \(U^HAU = \left[ \begin{array}{rr} -1 & 0 \\ 0 & 6 \end{array}\right]\)
3. \(U = \frac{1}{\sqrt{3}}\left[ \begin{array}{cc} 1 + i & 1 \\ -1 & 1 - i \end{array}\right]\), \(U^HAU = \left[ \begin{array}{rr} 1 & 0 \\ 0 & 4 \end{array}\right]\)
4. \(U = \frac{1}{\sqrt{3}}\left[ \begin{array}{ccc} \sqrt{3} & 0 & 0 \\ 0 & 1 + i & 1 \\ 0 & -1 & 1 - i \end{array}\right]\), \(U^HAU = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 3 \end{array}\right]\)

Show that \(\langle A \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{x}, A^{H}\mathbf{y}\rangle\) holds for all \(n \times n\) matrices \(A\) and for all \(n\)-tuples \(\mathbf{x}\) and \(\mathbf{y}\) in \(\mathbb{C}^n\).

[ex:8_6_10]

1. Prove (1) and (2) of Theorem [thm:025575].
2. Prove Theorem [thm:025616].
3. Prove Theorem [thm:025659].

2. \(\| \lambda Z \|^2 = \langle \lambda Z, \lambda Z \rangle = \lambda\overline{\lambda} \langle Z, Z \rangle = |\lambda|^2 \| Z \|^2\)

1. Show that \(A\) is hermitian if and only if \(\overline{A} = A^T\).
2. Show that the diagonal entries of any hermitian matrix are real.

2. If the \((k, k)\)-entry of \(A\) is \(a_{kk}\), then the \((k, k)\)-entry of \(\overline{A}\) is \(\overline{a}_{kk}\) so the \((k, k)\)-entry of \((\overline{A})^T = A^{H}\) is \(\overline{a}_{kk}\). This equals \(a\), so \(a_{kk}\) is real.

1. Show that every complex matrix \(Z\) can be written uniquely in the form \(Z = A + iB\), where \(A\) and \(B\) are real matrices.
2. If \(Z = A + iB\) as in (a), show that \(Z\) is hermitian if and only if \(A\) is symmetric, and \(B\) is skew-symmetric (that is, \(B^{T} = -B\)).

If \(Z\) is any complex \(n \times n\) matrix, show that \(ZZ^{H}\) and \(Z + Z^{H}\) are hermitian.

A complex matrix \(B\) is called skew-hermitian if \(B^{H} = -B\).

1. Show that \(Z - Z^{H}\) is skew-hermitian for any square complex matrix \(Z\).
2. If \(B\) is skew-hermitian, show that \(B^{2}\) and \(iB\) are hermitian.
3. If \(B\) is skew-hermitian, show that the eigenvalues of \(B\) are pure imaginary (\(i \lambda\) for real \(\lambda\)).
4. Show that every \(n \times n\) complex matrix \(Z\) can be written uniquely as \(Z = A + B\), where \(A\) is hermitian and \(B\) is skew-hermitian.

2. Show that \((B^2)^H = B^HB^H = (-B)(-B) = B^2\); \((iB)^H = \overline{i}B^H = (-i)(-B) = iB\).
3. If \(Z = A + B\), as given, first show that \(Z^{H} = A - B\), and hence that \(A = \frac{1}{2}(Z + Z^{H})\) and \(B = \frac{1}{2}(Z - Z^{H})\).

Let \(U\) be a unitary matrix. Show that:

1. \(\| U\mathbf{x} \| = \| \mathbf{x} \|\) for all columns \(\mathbf{x}\) in \(\mathbb{C}^n\).
2. \(|\lambda| = 1\) for every eigenvalue \(\lambda\) of \(U\).

1. If \(Z\) is an invertible complex matrix, show that \(Z^{H}\) is invertible and that \((Z^{H})^{-1} = (Z^{-1})^{H}\).
2. Show that the inverse of a unitary matrix is again unitary.
3. If \(U\) is unitary, show that \(U^{H}\) is unitary.

2. If \(U\) is unitary, \((U^{-1})^{-1} = (U^{H})^{-1} = (U^{-1})^{H}\), so \(U^{-1}\) is unitary.

Let \(Z\) be an \(m \times n\) matrix such that \(Z^{H}Z = I_{n}\) (for example, \(Z\) is a unit column in \(\mathbb{C}^n\)).

1. Show that \(V = ZZ^{H}\) is hermitian and satisfies
   \(V^{2} = V\).
2. Show that \(U = I - 2ZZ^{H}\) is both unitary and hermitian (so \(U^{-1} = U^{H} = U\)).

1. If \(N\) is normal, show that \(zN\) is also normal for all complex numbers \(z\).
2. Show that (a) fails if normal is replaced by hermitian.

2. \(H = \left[ \begin{array}{rr} 1 & i \\ -i & 0 \end{array}\right]\) is hermitian but \(iH = \left[ \begin{array}{rr} i & -1 \\ 1 & 0 \end{array}\right]\) is not.

Show that a real \(2 \times 2\) normal matrix is either symmetric or has the form \(\left[ \begin{array}{rr} a & b \\ -b & a \end{array}\right]\).

If \(A\) is hermitian, show that all the coefficients of \(c_{A}(x)\) are real numbers.

1. If \(A = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right]\), show that \(U^{-1}AU\) is not diagonal for any invertible complex matrix \(U\).
2. If \(A = \left[ \begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right]\), show that \(U^{-1}AU\) is not upper triangular for any real invertible matrix \(U\).

2. Let \(U = \left[ \begin{array}{rr} a & b \\ c & d \end{array}\right]\) be real and invertible, and assume that \(U^{-1}AU = \left[ \begin{array}{rr} \lambda & \mu \\ 0 & v \end{array}\right]\). Then \(AU = U\left[ \begin{array}{rr} \lambda & \mu \\ 0 & v \end{array}\right]\), and first column entries are \(c = a\lambda\) and \(-a = c\lambda\). Hence \(\lambda\) is real (\(c\) and \(a\) are both real and are not both \(0\)), and \((1 + \lambda^{2})a = 0\). Thus \(a = 0\), \(c = a\lambda = 0\), a contradiction.

If \(A\) is any \(n \times n\) matrix, show that \(U^{H}AU\) is lower triangular for some unitary matrix \(U\).

If \(A\) is a \(3 \times 3\) matrix, show that \(A^{2} = 0\) if and only if there exists a unitary matrix \(U\) such that \(U^{H}AU\) has the form \(\left[ \begin{array}{rrr} 0 & 0 & u \\ 0 & 0 & v \\ 0 & 0 & 0 \end{array}\right]\) or the form \(\left[ \begin{array}{rrr} 0 & u & v \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\).

If \(A^{2} = A\), show that rank \(A = \func{tr}A\). [Hint: Use Schur’s theorem.]

Let \(A\) be any \(n \times n\) complex matrix with eigenvalues \(\lambda_1, \dots, \lambda_n\). Show that \(A = P+N\) where \(N^{n}=0\) and \(P=UDU^{T}\) where \(U\) is unitary and \(D=\func{diag}(\lambda_1,\dots,\lambda_{n})\). [Hint: Schur’s theorem]