## 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 $S$.

## 每週問題 August 8, 2016

For a set of vectors $S=\{\mathbf{v}_1,\ldots,\mathbf{v}_n\}$, prove that $\hbox{span}(S)$ is the intersection of all subspaces that contain $S$.

## 每週問題 August 1, 2016

For the matrix

$A=\begin{bmatrix} 1&0&0&1/3&1/3&1/3\\ 0&1&0&1/3&1/3&1/3\\ 0&0&1&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3 \end{bmatrix}$,

determine $A^{300}$.

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## 每週問題 July 25, 2016

Suppose that $A$ and $B$ are $m\times n$ complex matrices. If $A\mathbf{x}=B\mathbf{x}$ holds for every $\mathbf{x}\in\mathbb{C}^n$, prove that $A=B$.

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