Math Problems Vault

Problem 24.2.1

(i)(ii)(iii)

Let $A$ , $B$ be $n \times n$ matrices. Assume that the entries of $A$ and $B$ are analytic functions of $x$ .

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(i)

Show that

\frac{d}{dx}(AB) = \left( \frac{d}{dx} A \right) B + A \left( \frac{d}{dx} B \right).

(ii)

Show that

\frac{d}{dx}(A \otimes B) = \left( \frac{d}{dx} A \right) \otimes B + A \otimes \left( \frac{d}{dx} B \right).

(iii)

Show that

\frac{d}{dx}(A \bullet B) = \left( \frac{d}{dx} A \right) \bullet B + A \bullet \left( \frac{d}{dx} B \right)

where $\bullet$ denotes the Hadamard product.

Problem 24.2.2

Let $A$ be an $m \times n$ matrix over $\mathbb {R}$ and

\overrightarrow {x} := (1, x, x^{2}, \ldots , x^{m-1})^{T}, \quad \overrightarrow {y} := (1, y, y^{2}, \ldots , y^{n-1})^{T}.

Find the extrema of the function $p(\overrightarrow {x}, \overrightarrow {y}) = \overrightarrow {x}^{T} A \overrightarrow {y}$ .

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Problem 24.2.3

(i) Let $\epsilon \in \mathbb {R}$ . Let $A(\epsilon )$ be an invertible $n \times n$ matrix. Assume that the entries $a_{jk}$ are analytic functions of $\epsilon$ . Show that

\operatorname {tr}\left( A^{-1}(\epsilon ) \frac{d}{d\epsilon } A(\epsilon ) \right) = \frac{1}{\operatorname {det}\left(A(\epsilon )\right)} \frac{d}{d\epsilon } \operatorname {det}\left(A(\epsilon )\right).

(ii) Consider the $2 \times 2$ matrix

M(\epsilon ) = \begin{pmatrix} f_{1}(\epsilon ) & f_{2}(\epsilon ) \\ f_{3}(\epsilon ) & f_{4}(\epsilon ) \end{pmatrix}

where $f_{j}$ ( $j = 1,2,3,4$ ) are smooth functions and $\operatorname {det}\left(M(\epsilon )\right) > 0$ for all $\epsilon$ . Show that

\operatorname {tr}((dM(\epsilon )) M(\epsilon )^{-1}) = d(\ln (\operatorname {det}\left(M(\epsilon )\right)))

where $d$ is the exterior derivative.

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Problem 24.2.4

Let $f_{j} : \mathbb {R} \rightarrow \mathbb {R}$ ( $j = 1, \ldots , n$ ) be analytic functions. Consider the determinant (Wronskian)

W(x) = \det \begin{pmatrix} f_{1}(x) & f_{2}(x) & \cdots & f_{n}(x) \\ f_{1}'(x) & f_{2}'(x) & \cdots & f_{n}'(x) \\ \vdots & \vdots & & \vdots \\ f_{1}^{(n-1)}(x) & f_{2}^{(n-1)}(x) & \cdots & f_{n}^{(n-1)}(x) \end{pmatrix}

where $f_{j}^{(n-1)}$ denotes the $(n-1)$ -th derivative of $f_{j}$ . If the Wronskian of these functions is not identically $0$ on $\mathbb {R}$ , then these functions form a linearly independent set. Let $n = 3$ and $f_{1}(x) = x$ , $f_{2}(x) = e^{x}$ , $f_{3}(x) = e^{2x}$ . Find the Wronskian. Discuss.

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Problem 24.2.5

Let $f_{1}, f_{2}, f_{3} : \mathbb {R}^{3} \rightarrow \mathbb {R}$ be continuously differentiable functions. Find the determinant of the $3 \times 3$ matrix $A = (a_{jk})$

a_{jk} := \frac{\partial f_{j}}{\partial x_{k}} - \frac{\partial f_{k}}{\partial x_{j}}.

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Problem 24.2.6

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Let $\overrightarrow {u} \in \mathbb {C}^{n}$ be considered as a column vector and

\partial := (\partial /\partial x_{1} \ \cdots \ \partial /\partial x_{n})^{T}.

Is

[\overrightarrow {u}^{*} A \partial , \overrightarrow {u}^{*} B \partial ] = \overrightarrow {u}^{*} [A, B] \partial \ ?

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Problem 24.2.7

Let $V(\alpha )$ be a $2 \times 2$ matrix where all the entries are smooth functions of $\alpha$ . Calculate $dV(\alpha )/d\alpha$ and then find the conditions on the entries such that $[dV(\alpha )/d\alpha , V(\alpha )] = 0_{2}$ .

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Problem 24.2.8

Let $j$ be a positive integer. Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {R}$ . Calculate

\lim _{\epsilon \rightarrow 0} \frac{1}{\epsilon } ((A + \epsilon B)^{j} - A^{j}), \quad \left. \frac{d}{d\epsilon } \operatorname {tr}(A + \epsilon B)^{j} \right|_{\epsilon = 0}.

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Problem 24.2.9

Find the partial differential equation given by the condition

\det \begin{pmatrix} 0 & \partial u/\partial x_{1} & \partial u/\partial x_{2} \\ \partial u/\partial x_{1} & \partial ^{2} u/\partial x_{1}^{2} & \partial ^{2} u/\partial x_{1} \partial x_{2} \\ \partial u/\partial x_{2} & \partial ^{2} u/\partial x_{2} \partial x_{1} & \partial ^{2} u/\partial x_{2}^{2} \end{pmatrix} = 0.

Find a nontrivial solution of the partial differential equation.

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Problem 24.2.10

(i)(ii)

Let $\phi : \mathbb {R} \rightarrow \mathbb {R}$ be an analytic function. Consider the matrices

A(t) = \begin{pmatrix} e^{i\phi (t)} & 1 \\ 1 & e^{i d\phi (t)/dt} \end{pmatrix}, \quad B(t) = \begin{pmatrix} 1 & e^{i\phi (t)} \\ e^{i d\phi (t)/dt} & 1 \end{pmatrix}.

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(i)

Find the differential equation for $\phi$ from the condition $\operatorname {tr}(AB) = 0$ .

(ii)

Find the differential equation for $\phi$ from the condition $\operatorname {det}\left(AB\right) = 0$ .

Problem 24.2.11

Consider the four $6 \times 6$ matrices

\beta _{0} = \begin{pmatrix} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{pmatrix}, \quad \beta _{1} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \end{pmatrix},

\beta _{2} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}, \quad \beta _{3} = \begin{pmatrix} 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}

and $\psi = (E_{1} \ E_{2} \ E_{3} \ cB_{1} \ cB_{2} \ cB_{3})^{T}$ . Show that

\beta _{0} \frac{\partial }{c \partial t} \psi + \beta _{1} \frac{\partial }{\partial x_{1}} \psi + \beta _{2} \frac{\partial }{\partial x_{2}} \psi + \beta _{3} \frac{\partial }{\partial x_{3}} \psi = (0 \ 0 \ 0 \ 0 \ 0 \ 0)^{T}

are Maxwell's equations

\frac{1}{c^{2}} \frac{\partial \overrightarrow {E}}{\partial t} = \nabla \times \overrightarrow {B}, \quad -\frac{\partial \overrightarrow {B}}{\partial t} = \nabla \times \overrightarrow {E}.

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Problem 24.2.12

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices and $f_{j}(x_{1}, x_{2})$ ( $j = 1,2,3$ ) be real-valued smooth functions $f_{j} : \mathbb {R}^{2} \rightarrow \mathbb {R}$ . Consider the $2 \times 2$ matrix

N(x_{1}, x_{2}) = f_{1} \sigma _{1} + f_{2} \sigma _{2} + \sigma _{3} f_{3} \equiv \begin{pmatrix} f_{3} & -if_{2} + f_{1} \\ if_{2} + f_{1} & -f_{3} \end{pmatrix}.

Find $dN$ , $N^{*}$ . Then calculate $d(N^{*} dN)$ . Find the conditions of $f_{1}$ , $f_{2}$ , $f_{3}$ such that $d(N^{*} dN) = 0_{2}$ .

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Problem 24.2.13

Let $f_{11}, f_{22}, f_{33}$ be analytic functions $f_{jj} : \mathbb {R} \rightarrow \mathbb {R}$ . Consider the $4 \times 4$ matrix

M(x) = \begin{pmatrix} f_{11}(x) & 0 & 0 & 0 \\ 1 & f_{22}(x) & 0 & 1 \\ 0 & 0 & f_{33}(x) & 1 \\ f_{11}'(x) & f_{22}'(x) & f_{33}'(x) & 0 \end{pmatrix}

where $'$ denotes differentiation with respect to $x$ . Find the determinant of the matrix and write down the ordinary differential equation which follows from $\operatorname {det}\left(M(x)\right) = 0$ . Find solutions of the differential equation.

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