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title: svd_1_team

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title: svd_1_team 
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**Section 1**

Example 1: Fill in the gaps 

$\begin{bmatrix} 1&1&1&1&1&1\\1&1&1&1&1&1\\1&1&1&1&1&1\\1&1&1&1&1&1\\1&1&1&1&1&1\\1&1&1&1&1&1\\\end{bmatrix}=\begin{bmatrix}1\\ 1\\ 1\\ 1\\ 1\\  1 \end{bmatrix}[~1\ ~1\ ~1\ ~1\ ~1\ ~1\ ]$

Example 2: Fill in the gaps

$\begin{bmatrix} a&a&c&c&e&e\\a&a&c&c&e&e\\a&a&c&c&e&e\\a&a&c&c&e&e\\a&a&c&c&e&e\\a&a&c&c&e&e\\\end{bmatrix}=\begin{bmatrix}1\\ 1\\ 1\\ 1\\ 1\\ 1\\  \end{bmatrix}[~a\ ~~a\ ~~c\  ~~c\ ~~e\  ~~e\         ]$

Example 3(a): Fill in the gaps

$\begin{bmatrix} 1&0\\1&1\\\end{bmatrix}=\begin{bmatrix}a\\ b\\   \end{bmatrix}[~c\ ~~\ d~   ]$

$ac=1,~ ad=0$. Then $c=d$. 
$bc=1, ~bd=1$. Then $c=d$.
Therefore, either $a$ or $d$ must be $0$ which is not possible if they must produce nonzero products in other elements.


Example 3(b): Fill in the gaps

$\begin{bmatrix} 1&0\\1&1\\\end{bmatrix}=\begin{bmatrix}1\\ 0\\   \end{bmatrix}[~1\ ~~0\ ~      ]+\begin{bmatrix}0\\ 1\\   \end{bmatrix}[~1\ ~~1\ ~]$

Draw some conclusions.

The null space of the matrix in #3 contains only the $\mathbf{0}$ vector.

Rows and columns must be linearly dependent.

**Section 2**

1. Find rank one 2 by 2  matrix $\tilde A$ closest to $A$. Use Frobenius norm. To solve the arising system of equations use
https://www.wolframalpha.com/input/?i=solve+system+of+equations

$\lVert A \rVert_F = 2  \sqrt{842}$

3. Then compute $E=A-\tilde A$, and write $A$ as a sum of $\tilde A$ and $E$.
$A=\begin{bmatrix} 30&28\\28&30\\\end{bmatrix}$

2. How are $\tilde A$ and $E$ are related to the eigenvalues and eigenvectors of $A$? Try to compute $\tilde A$ using eigenvalues and eigenvectors. This is a famous kind of decomposition, called how? 
Can this decomposition be obtained for any matrix?
E.g. Example 3.

**Section 3**

Use Wolphram Alpha for all computations

1. Find rank one 2 by 2  matrix $\tilde A$ closest to $A$. Use Frobenius norm. 
$\tilde A=\begin{bmatrix} 58&0\\0&2\\\end{bmatrix}$

2. Compute the eigenvalues and the unit eigenvectors of $AA^T$ and $A^TA$. What do you notice about the eigenvalues of $A^TA$ and $AA^T$?

$A^TA = A A^T = \begin{bmatrix}  1684 & 1680 \\ 1680 & 1684\end{bmatrix}$



3. Let $\lambda_1$ be the larget eigenvalue, and let ${\bf u_1}$ be the unit eigenvector of $AA^T$ corresponding to $\lambda_1$, let ${\bf v_1}$ be the unit eigenvector of $A^TA$ corresponding to $\lambda_1$. Let $\sigma_1$ be the square root of $\lambda_1$. Compute 
 $$A_1=\sigma_1{\bf u_1}{\bf v_1}^T$$
 
 $A_1=58\begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2}\end{bmatrix}$
 
 4. Compare $\tilde A$ and $A_1$
 
 5. Prove that for any non-zero vectors ${\bf u}$ and ${\bf v}$, the matrix ${\bf u}{\bf v}^T$ has rank 1. 

6. Suppose you have an $m\times n$ matrix $M$. Suggest a linear algebra procedure for obtaining the matrix $M_1$, which is the closest rank 1 matrix to $M$ in Frobenius norm.