## 每週問題 October 17, 2016

Let $\mathbf{a}_1,\ldots,\mathbf{a}_n$ be $n$ vectors in $\mathbb{R}^m$. Show that $m\times m$ Gramian matrix $G=\sum_{i=1}^n\mathbf{a}_i\mathbf{a}_i^T$ is nonsingular if and only if $\hbox{span}\{\mathbf{a}_1,\ldots,\mathbf{a}_n\}=\mathbb{R}^m$.

## 每週問題 October 10, 2016

Let $A=[a_{ij}]$ be an $n\times n$ Hermitian matrix. If $A$ is positive semi-definite and $a_{ii}=0$ for some $i$, show that $a_{ij}=a_{ji}=0$ for all $j$.

## 每週問題 October 3, 2016

Let $A$ and $B$ be $n\times n$ Hermitian matrices. Suppose $A$ is invertible. Show that there exists a nonsingular matrix $P$ so that $P^\ast AP$ and $P^\ast BP$ are diagonal if and only if $A^{-1}B$ is diagonalizable and all its eigenvalues are real.

## 每週問題 September 26, 2016

Let $A$ be a diagonalizable matrix with real eigenvalues. Show that $A$ can be represented as $A=BC$, where $B$ is a positive definite matrix and $C$ is a Hermitian matrix.

## 每週問題 September 19, 2016

Let $A$ be a real symmetric positive semi-definite matrix. If $\mathbf{x}^TA\mathbf{x}=0$, show that $A\mathbf{x}=\mathbf{0}$.

## 每週問題 September 12, 2016

Let $A$ and $B$ be $n\times n$ matrices. Suppose that the eigenvectors of $A$ span $\mathbb{C}^n$ and have distinct eigenvalues. Show that $AB=BA$ if and only if $A$ and $B$ have the same set of eigenvectors (with possibly different eigenvalues).

## 每週問題 September 5, 2016

Let $A$ be an $m\times n$ real matrix and $\mathbf{b}\in\mathbb{R}^m$. Solve the Tikhonov regularization problem:

$\displaystyle \min_{\mathbf{x}}\Vert A\mathbf{x}-\mathbf{b}\Vert_2^2+\lambda\Vert \mathbf{x}\Vert_2^2$,

where $\lambda>0$.

## 每週問題 August 29, 2016

Let $A$ be an $m\times m$ matrix, $B$ be an $n\times m$ matrix and $C$ be an $n\times n$ matrix. If $A$ and $C$ are symmetric positive definite, show the following identities.
(a) $(A^{-1}+B^TC^{-1}B)^{-1}B^TC^{-1}=AB^T(BAB^T+C)^{-1}$
(b) $(A^{-1}+B^TC^{-1}B)^{-1}=A-AB^T(BAB^T+C)^{-1}BA$

## 每週問題 August 22, 2016

Let $\mathcal{V}$ and $\mathcal{W}$ be finite dimensional vector spaces, and $T:\mathcal{V}\to\mathcal{W}$ be a linear transformation. For a subspace $\mathcal{X}$ of $\mathcal{V}$, the image $T(\mathcal{X})=\{T(\mathbf{x})|\mathbf{x}\in\mathcal{X}\}$ of $\mathcal{X}$ under $T$ is a subspace of $\mathcal{W}$. Prove that if $\mathcal{X}\cap N(T)=\{\mathbf{0}\}$, then $\dim T(\mathcal{X})=\dim\mathcal{X}$. Note that $N(T)$ denotes the nullspace (kernel) of $T$.

## 每週問題 August 15, 2016

Let $S=\{\mathbf{0}\}$ be the set containing only the zero vector.
(a) Explain why $S$ must be linearly dependent.
(b) Explain why the empty set is a basis for $S$.