Chapter Problems

(i) Consider the $2 \times 2$ rotation matrix

with $\operatorname {det}\left(R(\alpha )\right) = 1$. Is the determinant preserved under repeated differentiation of $R(\alpha )$ with respect to $\alpha$? (ii) Consider the $2 \times 2$ matrix

with $\operatorname {det}\left(S(\alpha )\right) = -1$. Is the determinant preserved under repeated differentiation of $S(\alpha )$ with respect to $\alpha$?

Let $f_{jk} : \mathbb {R} \rightarrow \mathbb {R}$ be analytic functions, where $j, k = 1, 2$ and $F(\epsilon ) = (f_{jk}(\epsilon ))$. Find the differential equations for $F(\epsilon )$ such that

Consider the invertible $2 \times 2$ matrix

Show that $d(\ln (\operatorname {det}\left(A(x)\right))) = \operatorname {tr}(A^{-1}(x) dA(x))$ where $d$ denotes the exterior derivative.

(i) Consider the analytic function $\overrightarrow {f} : \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}$

Show that this function admits the (only) fixed point $(0,0)$. Find the functional matrix at the fixed point

(ii) Consider the analytic function $\overrightarrow {g} : \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}$

Show that this function admits the (only) fixed point $(0,0)$. Find the functional matrix at the fixed point

(iii) Multiply the two matrices found in (i) and (ii). (iv) Find the composite function $\overrightarrow {h} : \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}$

Show that this function also admits the fixed point $(0,0)$. Find the functional matrix at this fixed point

Compare this matrix with the matrix found in (iii).

Consider the $2 \times 2$ matrix

Calculate $dV(t)/dt$ and then find the commutator $[dV(t)/dt, V(t)]$. Discuss.

Let $A$ be an $n \times n$ matrix. Assume that the inverse of $A$ exists, i.e. $\operatorname {det}\left(A\right) \neq 0$. Then the inverse $B = A^{-1}$ can be calculated as

Apply this formula to the $2 \times 2$ matrix $A$ with $\operatorname {det}\left(A\right) = a_{11} a_{22} - a_{12} a_{21} \neq 0$.

Let $Q$ and $P$ be $n \times n$ symmetric matrices over $\mathbb {R}$, i.e. $Q = Q^{T}$ and $P = P^{T}$. Assume that $P^{-1}$ exists. Find the maximum of the function $f : \mathbb {R}^{n} \rightarrow \mathbb {R}$

subject to $\overrightarrow {x}^{T} P \overrightarrow {x} = 1$. Use the Lagrange multiplier method.

Let $\Phi : \mathbb {R}^{n} \times \mathbb {R}^{n} \rightarrow \mathbb {R}$ be an analytic function, where $(\overrightarrow {x}, \overrightarrow {y}) = (x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}) \in \mathbb {R}^{n} \times \mathbb {R}^{n}$. The Monge-Ampere determinant $M(\Phi )$ is defined by

Let $n = 2$ and

Find the Monge-Ampere determinant and the conditions on $x_{1}, x_{2}, y_{1}, y_{2}$ such that $M(\Phi ) = 0$.

Consider the $m \times m$ matrix $F(\overrightarrow {x}) = (f_{jk}(\overrightarrow {x}))$ ($j, k = 1, \ldots , m$), where $f_{jk} : \mathbb {R}^{n} \rightarrow \mathbb {R}$ are analytic functions. Assume that $F(\overrightarrow {x})$ is invertible for all $\overrightarrow {x} \in \mathbb {R}^{n}$. Then we have the identities ($j = 1, \ldots , m$)

and

The differentiation is understood entrywise. Apply the identities to the matrix ($m = 2, n = 1$)

Let $A$ be an invertible $n \times n$ matrix over $\mathbb {R}$. Consider the functions

where $j = 1, \ldots , n$, $\overrightarrow {c}_{j}$ is the $j$-th column of the inverse matrix of $A$, $\overrightarrow {e}_{j}$ is the $j$-th column of the $n \times n$ identity matrix. This means $\overrightarrow {e}_{1}, \ldots , \overrightarrow {e}_{n}$ is the standard basis (as column vectors) in $\mathbb {R}^{n}$. The $\overrightarrow {c}_{j}$ are determined by minimizing the $E_{j}$ with respect to the $\overrightarrow {c}_{j}$. Apply this method to find the inverse of the $3 \times 3$ matrix

Let $\overrightarrow {f}$ be a function from $U$, an open subset of $\mathbb {R}^{m}$, to $\mathbb {R}^{n}$. Assume that the component function $f_{j}$ ($j = 1, \ldots , n$) possess first order partial derivatives. Then we can associate the $n \times m$ matrix

where $\overrightarrow {p} \in U$. The matrix is called the Jacobian matrix of $\overrightarrow {f}$ at the point $\overrightarrow {p}$. When $m = n$ the determinant of the square matrix $\overrightarrow {f}$ is called the Jacobian of $\overrightarrow {f}$. Let

and $\overrightarrow {f} : A \times B \rightarrow \mathbb {R}^{2}$ with $f_{1}(r, \theta ) = r \cos (\theta )$, $f_{2}(r, \theta ) = r \sin (\theta )$. Find the Jacobian matrix and the Jacobian.

Consider the rotation matrix

where $\omega$ is a fixed frequency. Find the $2 \times 2$ matrix

and show it is hermitian.

Let $GL(m, \mathbb {C})$ be the general linear group over $\mathbb {C}$. This Lie group consists of all nonsingular $m \times m$ matrices. Let $G$ be a Lie subgroup of $GL(m, \mathbb {C})$. Suppose $u_{1}, u_{2}, \ldots , u_{n}$ is a coordinate system on $G$ in some neighborhood of $I_{m}$, the $m \times m$ identity matrix, and that $X(u_{1}, u_{2}, \ldots , u_{m})$ is a point in this neighborhood. The matrix $dX$ of differential one-forms contains $n$ linearly independent differential one-forms since the $n$-dimensional Lie group $G$ is smoothly embedded in $GL(m, \mathbb {C})$. Consider the matrix of differential one forms

The matrix $\Omega$ of differential one forms contains $n$-linearly independent ones.

Consider $GL(m, \mathbb {R})$ and a Lie subgroup of it. We interpret each element $X$ of $G$ as a linear transformation on the vector space $\mathbb {R}^{m}$ of row vectors $\overrightarrow {v} = (v_{1}, v_{2}, \ldots , v_{n})$. Thus $\overrightarrow {v} \rightarrow \overrightarrow {w} = \overrightarrow {v} X$. Show that $d\overrightarrow {w} = \overrightarrow {w} \Omega$.

Consider the compact Lie group $SO(2)$ consisting of the matrices

Calculate $dX$ and $X^{-1} dX$.

Let $n$ be the dimension of the Lie group $G$. Since the vector space of differential one-forms at the identity element is an $n$-dimensional vector space, there are exactly $n$ linearly independent left invariant differential one-forms in $G$. Let $\sigma_{1}, \sigma_{2}, \ldots , \sigma_{n}$ be such a system. Consider the Lie group

Let

Consider the Lie group consisting of the matrices

Calculate $X^{-1}$ and $X^{-1} dX$. Find the left-invariant differential one-forms and the left-invariant volume element.

Let $A$ be a $2 \times 2$ symmetric matrix

over $\mathbb {R}$. We define

Show that

Consider the matrix

Find the function (characteristic polynomial) $p(\lambda ) = \operatorname {det}\left(A - \lambda I_{2}\right)$. Find the eigenvalues of $A$ by solving $p(\lambda ) = 0$. Find the minima of the function $f(\lambda ) = \left|p(\lambda )\right|$. Discuss.

Let $\alpha , \beta \in \mathbb {R}$. Consider the $2 \times 2$ matrix

Find $\exp (tB)$, where $t \in \mathbb {R}$ and thus solve the initial value problem of the matrix differential equation

The Fréchet derivative of a matrix function $f : \mathbb {C}^{n \times n}$ at a point $X \in \mathbb {C}^{n \times n}$ is a linear mapping $L_{X} : \mathbb {C}^{n \times n} \rightarrow \mathbb {C}^{n \times n}$ such that for all $Y \in \mathbb {C}^{n \times n}$

Calculate the Fréchet derivative of $f(X) = X^{2}$.