Supplementary Problems

Show that the Hadamard product is linear. Show that the Hadamard product is associative.

Let $A$, $B$ be $n \times n$ positive semidefinite matrices. Show that $A \bullet B$ is also positive semidefinite.

Let $A$ be a positive semidefinite $n \times n$ matrix. Let $B$ be an $n \times n$ matrix with $\left\| B\right\| \leq 1$, where $\left\| \cdot \right\|$ denotes the spectral norm. Show that

where $\left\| \cdot \right\|$ denotes the spectral norm and $j = 1, \ldots , n$.

Let $A$, $B$ be $m \times n$ matrices and $D$ and $E$ be $m \times m$ and $n \times n$ diagonal matrices, respectively. Show that

Let $A$ be an $n \times n$ diagonalizable matrix over $\mathbb {C}$ with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ and diagonalization $A = TDT^{-1}$, where $D$ is a diagonal matrix such that $D_{jj} = \lambda_{j}$ for all $j = 1, \ldots , n$. Show that

Show that $\operatorname {tr}(A(B \bullet C)) \equiv (\operatorname {vec}(A^{T} \bullet B))^{T}\operatorname {vec}(C)$. The left-hand side yields

Consider the $n \times n$ matrices with entries $\pm 1$. If $A$, $B$ such matrices, then $A \bullet B$ is again such a matrix. Do these matrices form a group under the Hadamard product? The neutral element would be the matrix $J$ with all entries $1$.

Consider the $n \times n$ matrices with entries $0$ and $1$. Let $\oplus$ be the XOR operation and we define $A \bullet B = (a_{ij} \oplus b_{ij})$. Then $A \bullet B$ is again a matrix with entries $0$ and $1$. Do these matrices form a group?