Chapter Problems

Consider the vectors in $\mathbb {C}^{4}$

Find the Euclidean norm and then normalize the vectors. Are the vectors orthogonal to each other?

Consider the vectors in $\mathbb {R}^{3}$

Let $\mathbf{A}$ be an $n \times n$ matrix over $\mathbb {R}$. The spectral norm is

It can be shown that $\left\| \mathbf{A}\right\|_{2}$ can also be calculated as

The eigenvalues of $\mathbf{A}^{T}\mathbf{A}$ are real and nonnegative. Consider the $2 \times 2$ matrix

Calculate $\left\| \mathbf{A}\right\|_{2}$ using this method.

Let $\left\{ \overrightarrow {v}_{j} : j = 1, \ldots , r\right\}$ be an orthogonal set of vectors in $\mathbb {R}^{n}$ with $r \leq n$. Show that

Consider the $2 \times 2$ matrix $\mathbf{A}$ over $\mathbb {C}$. Find the norm of $\mathbf{A}$ implied by the scalar product $\left\langle \mathbf{A}, \mathbf{A} \right\rangle = \sqrt{\operatorname {tr}(\mathbf{A}\mathbf{A}^{*})}$.

Let $\mathbf{A}$, $\mathbf{B}$ be $n \times n$ matrices over $\mathbb {C}$. A scalar product can be defined as

The scalar product implies a norm $\left\| \mathbf{A}\right\|^{2} = \left\langle \mathbf{A}, \mathbf{A} \right\rangle = \operatorname {tr}(\mathbf{A}\mathbf{A}^{*})$. This norm is called the Hilbert-Schmidt norm.

Find the norm $\left\| \mathbf{A}\right\| = \sqrt{\operatorname {tr}(\mathbf{A}^{*}\mathbf{A})}$ of the skew-hermitian matrix

without calculating $\mathbf{A}^{*}$.

Let $\mathbf{A}$ and $\mathbf{B}$ be $2 \times 2$ diagonal matrices over $\mathbb {R}$. Assume that

and

Can we conclude that $\mathbf{A} = \mathbf{B}$?

Let $\mathbf{A}$ be an $n \times n$ matrix over $\mathbb {C}$. Let $\left\| \cdot \right\|$ be a subordinate matrix norm for which $\left\| I_{n}\right\| = 1$. Assume that $\left\| \mathbf{A}\right\| < 1$.

Let $\mathbf{A}$ be an $n \times n$ matrix. Assume that $\left\| \mathbf{A}\right\| < 1$. Show that

Let $\mathbf{A}$ be an $n \times n$ nonsingular matrix and $\mathbf{B}$ an $n \times n$ matrix. Assume that $\left\| \mathbf{A}^{-1}\mathbf{B}\right\| < 1$.

Let $\mathbf{A}$, $\mathbf{B}$ be $n \times n$ matrices over $\mathbb {R}$ and $t \in \mathbb {R}$. Let $\left\| \cdot \right\|$ be a matrix norm. Show that

Let $\mathbf{A}_{1}, \mathbf{A}_{2}, \ldots , \mathbf{A}_{p}$ be $m \times m$ matrices over $\mathbb {C}$. Then we have the inequality

and (Trotter formula)

Let $p = 2$. Find the estimate for the $2 \times 2$ matrices

Let $U_{1}$, $U_{2}$ be unitary $n \times n$ matrices. Let $\overrightarrow {v}$ be a normalized vector in $\mathbb {C}^{n}$. Consider the norm of a $k \times k$ matrix $M$

where $\left\| \overrightarrow {v}\right\|$ denotes the Euclidean norm. Show that if $\left\| U_{1}-U_{2}\right\| \leq \epsilon$ then $\left\| U_{1}\overrightarrow {v}-U_{2}\overrightarrow {v}\right\| \leq \epsilon$.

Given the $2 \times 2$ matrix (rotation matrix)

Calculate $\left\| R(\alpha )\right\| = \sup_{\left\| \overrightarrow {x}\right\| =1}\left\| R(\alpha )\overrightarrow {x}\right\|$.

Let $\mathbf{A}$ be an $n \times n$ matrix. Let $\rho (\mathbf{A})$ be the spectral radius of $\mathbf{A}$. Then we have

Let

Calculate $\rho (\mathbf{A})$ and the right-hand side of the inequality.

Let $t \in \mathbb {R}$. Consider the symmetric $3 \times 3$ matrix over $\mathbb {R}$

Find the condition on $t$ such that $\rho (\mathbf{A}(t)) < 1$, where $\rho (\mathbf{A}(t))$ denotes the spectral radius of $\mathbf{A}(t)$.

Let $\mathbf{A}$ be an $n \times n$ matrix. One approach to calculate $\mathrm{exp}\left(\mathbf{A}\right)$ is to compute an eigenvalue decomposition $\mathbf{A} = XBX^{-1}$ and then apply the formula $e^{\mathbf{A}} = Xe^{B}X^{-1}$. We have using the Schur decomposition

where $U$ is unitary, the matrix $N = (n_{jk})$ is a strictly upper triangular ($n_{jk} = 0, j \geq k$) and $\lambda (\mathbf{A}) = \left\{ \lambda_{1}, \ldots , \lambda_{n}\right\}$ is the spectrum of $\mathbf{A}$. Using the Padé approximation to calculate $e^{\mathbf{A}}$ we have

where

Consider the nonnormal matrix

Calculate $\left\| R_{11}-e^{\mathbf{A}}\right\|_{2}$, where $\left\| \cdot \right\|_{2}$ denotes the 2-norm.

Let $\mathbf{A}$ be an $n \times n$ matrix with $\left\| \mathbf{A}\right\| < 1$. Then $\ln (I_{n}+\mathbf{A})$ exists. Show that

Let $\mathbf{A}$, $\mathbf{B}$ be $n \times n$ matrices over $\mathbb {C}$. Show that $\left\| [\mathbf{A},\mathbf{B}]\right\| \leq 2\left\| \mathbf{A}\right\| \left\| \mathbf{B}\right\|$.

Let $\mathbf{A}$, $\mathbf{B}$ be $n \times n$ matrices over $\mathbb {C}$. Then

Let

Calculate $\left\| [\mathbf{A},\mathbf{B}]\right\|$, $\left\| \mathbf{A}\mathbf{B}\right\|$, $\left\| \mathbf{B}\mathbf{A}\right\|$, $\left\| \mathbf{A}\right\|$, $\left\| \mathbf{B}\right\|$ and thus verify the inequality for these matrices. The norm is given by $\left\| \mathbf{C}\right\| = \sqrt{\operatorname {tr}(\mathbf{C}\mathbf{C}^{*})}$.

Denote by $\left\| \cdot \right\|_{HS}$ the Hilbert-Schmidt norm and by $\left\| \cdot \right\|_{O}$ the operator norm, i.e.

where $\mathbf{A}$ is an $m \times n$ matrix over $\mathbb {C}$, $\overrightarrow {x} \in \mathbb {C}^{n}$ and $\left\| \overrightarrow {x}\right\|$ is the Euclidean norm.

Let $\mathbf{A}$ be an $n \times n$ matrix. The logarithmic norm is defined by

Let $\left\| \mathbf{A}\right\| := \sup_{\overrightarrow {x}=1}\left|\mathbf{A}\overrightarrow {x}\right|$. Let $\mathbf{A}$ be the $n \times n$ identity matrix $I_{n}$. Find $\mu [I_{n}]$.

Consider the Hilbert space $\mathbb {R}^{n}$ and $\overrightarrow {x}, \overrightarrow {y} \in \mathbb {R}^{n}$. The scalar product $\left\langle \overrightarrow {x}, \overrightarrow {y} \right\rangle$ is given by

Thus the norm is given by $\left\| \overrightarrow {x}\right\| = \sqrt{ \left\langle \overrightarrow {x}, \overrightarrow {x} \right\rangle }$. Show that $\left|\overrightarrow {x}^{T}\overrightarrow {y}\right| \leq \left\| \overrightarrow {x}\right\| \cdot \left\| \overrightarrow {y}\right\|$.

Let $\mathbf{A}$ be an $n \times n$ hermitian matrix. Let $\overrightarrow {u}, \overrightarrow {v} \in \mathbb {C}^{n}$ and $\lambda \in \mathbb {C}$. Consider the equation

Let $M$ be an $m \times n$ matrix over $\mathbb {C}$. The Frobenius norm of $M$ is given by

Let $U_{m}$ be $m \times m$ unitary matrix and $U_{n}$ be an $n \times n$ unitary matrix. Show that

Show that $\left\| M\right\|_{F}$ is the square root of the sum of the squares of the singular values of $M$.

Let $M$ be an $m \times n$ matrix over $\mathbb {C}$. Find the rank-1 $m \times n$ matrix $\mathbf{A}$ over $\mathbb {C}$ which minimizes

Apply the method to the $3 \times 2$ matrix

Let $\mathbf{A}$ be an $n \times n$ matrix over $\mathbb {C}$. The spectral radius of the matrix $\mathbf{A}$ is the nonnegative number $\rho (\mathbf{A})$ defined by

where $\lambda_{j}(\mathbf{A})$ ($j = 1, \ldots , n$) are the eigenvalues of $\mathbf{A}$. We define the norm of $\mathbf{A}$ as

where $\left\| \mathbf{A}\overrightarrow {x}\right\|$ denotes the Euclidean norm. Is $\rho (\mathbf{A}) \leq \left\| \mathbf{A}\right\|$? Prove or disprove.

Let $\mathbf{A}$ be an $n \times n$ matrix over $\mathbb {C}$ and $I_{m}$ be the $m \times m$ unit matrix. Consider the norm of a $k \times k$ matrix $M$

where $\left\| \overrightarrow {x}\right\|$ denotes the Euclidean norm. Show that $\left\| \mathbf{A} \otimes I_{m}\right\| = \left\| \mathbf{A}\right\|$.

Find $2 \times 2$ matrices $\mathbf{A}$ and $\mathbf{B}$ over $\mathbb {C}$ which minimize

and determine the minimum.

Consider the $4 \times 4$ matrix (Hamilton operator)

where $\omega$ is the frequency and $\hbar$ is the Planck constant divided by $2\pi$. Find the norm of $\hat{H}$, i.e.

applying two different methods. In the first method apply the Lagrange multiplier method, where the constraint is $\left\| \overrightarrow {x}\right\| = 1$. In the second method we calculate $\hat{H}^{*}\hat{H}$ and find the square root of the largest eigenvalue. This is then $\left\| \hat{H}\right\|$. Note that $\hat{H}^{*}\hat{H}$ is positive semi-definite.

Let $\mathbf{A}$ be an invertible $n \times n$ matrix over $\mathbb {R}$. Consider the linear system $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$. The condition number of $\mathbf{A}$ is defined as

Find the condition number for the matrix

for the infinity norm, 1-norm and 2-norm.

Let $\mathbf{A}$, $\mathbf{B}$ be nonsingular $n \times n$ matrices. Show that

for all matrix norms, where $\operatorname {Cond}$ denotes the condition number.