Chapter Problems

Let

Calculate $A \bullet B$, $A \bullet A^{-1}$, $A \bullet J$, $J \bullet B$.

Let $n \in \mathbb {N}$ and

Thus $\operatorname {det}\left(A\right) = \operatorname {det}\left(A^{-1}\right) = 1$. Find $\operatorname {det}\left(A \bullet A^{-1}\right)$.

Let $A$, $B$ be $m \times n$ matrices. Show that $\operatorname {rk}\left(A \bullet B\right) \leq \left(\operatorname {rk}\left(A\right)\right)\left(\operatorname {rk}\left(B\right)\right)$.

Let

be symmetric matrices over $\mathbb {R}$. Assume that $A$ and $B$ are positive definite. Show that $A \bullet B$ is positive definite using the trace and determinant.

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$. The spectral radius $\rho (A)$ is the radius of the smallest circle in the complex plane that contains all its eigenvalues. Every characteristic polynomial has at least one root. Let $A$, $B$ be nonnegative matrices. Then $\rho (A \bullet B) \leq \rho (A)\rho (B)$. Apply this inequality to the nonnegative matrices

There exists an $n^{2} \times n$ selection matrix $J$ such that $A \bullet B = J^{T}(A \otimes B)J$, where $J^{T}$ is defined as the $n \times n^{2}$ matrix $\begin{bmatrix} E_{11} & E_{22} & \ldots & E_{nn} \end{bmatrix}$ with $E_{ii}$ the $n \times n$ matrix of zeros except for a $1$ in the $(i,i)$-th position (elementary matrices). Prove this identity for the special case $n = 2$.

If $A, B$ are $n \times n$ positive definite matrices and $(a_{jj})$ are the diagonal entries of $A$, then

with equality if and only if $A$ is a diagonal matrix. Let

First show that $A$ and $B$ are positive definite and then calculate the left- and right-hand side of $(1)$.

Consider the symmetric $4 \times 4$ matrices

Calculate the Hadamard product $A \bullet B$. Show that $\left\| A \bullet B\right\| \leq \left\| A^{*}A\right\| \left\| B^{*}B\right\|$, where the norm is given by the Hilbert-Schmidt norm.

Let $A$, $B$, $C$ and $D^{T}$ be $n \times n$ matrices over $\mathbb {R}$. Show that

Let $V$, $W$ be two matrices of the same order. If all entries of $V$ are nonzero, then we say that $X$ is Schur invertible and define its Schur inverse, $V^{(-)}$, by $V^{(-)} \bullet V = J$, where $J$ is the matrix with all $1$'s.

The vector space $M_{n}(F)$ of $n \times n$ matrices acts on itself in three distinct ways: if $C \in M_{n}(F)$ we can define endomorphisms $X_{C}$, $\Delta_{C}$ and $Y_{C}$ by

Let $A$, $B$ be $n \times n$ matrices. Assume that $X_{A}$ is invertible and $\Delta_{B}$ is invertible in the sense of Schur. Note that $X_{A}$ is invertible if and only if $A$ is, and $\Delta_{B}$ is invertible if and only if the Schur inverse $B^{(-)}$ is defined. We say that $(A, B)$ is a one-sided Jones pair if

We call this the braid relation. Give an example for a one-sided Jones pair.

Let $A$, $B$ be $n \times n$ matrices. Let $\overrightarrow {e}_{1}, \ldots , \overrightarrow {e}_{n}$ be the standard basis vectors in $\mathbb {C}^{n}$. We form the $n^{2}$ column vectors $(A\overrightarrow {e}_{j}) \bullet (B\overrightarrow {e}_{k})$, $j, k = 1, \ldots , n$. If $A$ is invertible and $B$ is Schur invertible, then for any $j$

is a basis of the vector space $\mathbb {C}^{n}$. Let

Find these bases for these matrices.

Let $B = (b_{jk})$ be a diagonalizable $n \times n$ matrix with eigenvalues $\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n}$. Thus, there is nonsingular $n \times n$ matrix $A$ such that

Show that

Thus the vector of eigenvalues of $B$ is transformed to the vector of its diagonal entries by the coefficient matrix $A \bullet (A^{-1})^{T}$.

Let $A$, $B$, $C$, $D$ be $n \times n$ matrices over $\mathbb {R}$. Let $\overrightarrow {s}^{T} = (1 \; \ldots \; 1)$ be a row vector in $\mathbb {R}^{n}$. Show that

where $\Gamma = (\gamma_{ij})$ is a diagonal matrix with $\gamma_{jj} = \sum_{i=1}^{n} a_{ij}b_{ij}$ and $j = 1, \ldots , n$.

Given two matrices $A$ and $B$ of the same size. If all entries of $A$ are nonzero, then we say that $A$ is Schur invertible and define its Schur inverse, $A^{(-)}$, by $A_{ij}^{(-)} := 1/A_{ij}$. Equivalently, we have $A^{(-)} \bullet A = J$, where $J$ is the matrix with all ones. An $n \times n$ matrix $W$ is a type-II matrix if

where $I_{n}$ is the $n \times n$ identity matrix. Find such a matrix for $n = 2$.

Let $A$, $B$, $C$, $D$ be $2 \times 2$ matrices and $\otimes$ be the Kronecker product. Is