gibson:teaching:fall-2016:iam961:hw2
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gibson:teaching:fall-2016:iam961:hw2 [2016/09/22 07:03] gibson created |
gibson:teaching:fall-2016:iam961:hw2 [2016/09/29 08:07] (current) gibson |
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| **Problem 1:** Prove that any real-valued matrix $A$ has a real-valued SVD, i.e. an SVD factorization | **Problem 1:** Prove that any real-valued matrix $A$ has a real-valued SVD, i.e. an SVD factorization | ||
| - | $A=U \Sigma V^\dagger$ where $U,\Sigma,$ and $V$ are real-valued matrices. | + | $A=U \Sigma V^\dagger$ where $U,\Sigma,$ and $V$ are real-valued matrices. Hint: Do not re-do the existence uniqueness proof we did in class with a restriction to real matrices. Start with a possibly complex SVD $A=U \Sigma V^*$ (where $U$ and $V$ are possibly complex but $\Sigma$ is necessarily real), and use this to show there must be real-valued unitary $U'$ and $V'$ that also form an SVD $A=U' \Sigma V'^* $. I suspect it will be helpful to express the SVDs as sums over columns of $U$ and $V$, i.e. $A = |
| + | \sum_{j=1}^p \sigma_j u_j v^*_j$, as shown in problem 3.) | ||
| **Problem 2:** Show that if $A$ is m x n and $B$ is p x n, the product $AB^*$ can be written as | **Problem 2:** Show that if $A$ is m x n and $B$ is p x n, the product $AB^*$ can be written as | ||
| Line 13: | Line 14: | ||
| of notation than any kind of deep proof. It should take about three lines. | of notation than any kind of deep proof. It should take about three lines. | ||
| - | **Problem 3:** Let $A$ be an m x n matrix with SVD $A = U\ \Sigma V$. By applying the results of problem 2 to the matrices $U$ and $\Sigma V^*$, show that | + | **Problem 3:** Let $A$ be an m x n matrix with SVD $A = U\ \Sigma V^*$. By applying the results of problem 2 to the matrices $U$ and $\Sigma V^*$, show that |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| Line 37: | Line 38: | ||
| 0.6571638901489170 0.7135254915624212 | 0.6571638901489170 0.7135254915624212 | ||
| x = | x = | ||
| - | 1.50000000000000 | + | 0.8 |
| - | 1.00000000000000 | + | 0.2 |
| </code> | </code> | ||
| (a) compute the SVD $A=U \Sigma V^\dagger$. | (a) compute the SVD $A=U \Sigma V^\dagger$. | ||
| Line 48: | Line 49: | ||
| (d) superimpose the image of that unit circle under $A$ on the $u_1, u_2$ plot. I.e. for $x \in C$ plot $Ax$. What can you say about the image of $C$ under $A$ in relation to the SVD? | (d) superimpose the image of that unit circle under $A$ on the $u_1, u_2$ plot. I.e. for $x \in C$ plot $Ax$. What can you say about the image of $C$ under $A$ in relation to the SVD? | ||
| - | (e) Plot the vector $x$ on the $v_1, v_2$ plot | + | (e) Plot the vector $x$ on the $v_1, v_2$ plot. |
| - | <code> | + | |
| - | x = | + | |
| - | 1.50000000000000 | + | |
| - | 1.00000000000000 | + | |
| - | </code> | + | |
| Now figure out where $y=Ax$ will be on the $u_1, u_2$ plot **geometrically** using the SVD, as follows | Now figure out where $y=Ax$ will be on the $u_1, u_2$ plot **geometrically** using the SVD, as follows | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| y = Ax = U \Sigma V^\dagger x = u_1 \sigma_1 v_1^\dagger x + u_2 \sigma_2 v_2^\dagger x | y = Ax = U \Sigma V^\dagger x = u_1 \sigma_1 v_1^\dagger x + u_2 \sigma_2 v_2^\dagger x | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Following this formula, measure (with a ruler!) the components of $x$ along $v_1$ and $v_2$. Multiply those lengths by the scaling factors $\sigma_1$ and $\sigma_2$. Then measure out the scaled lengths along $u_1$ and $u_2$, and add those components together to get the position of the vector $y=Ax$. Do this all by eye or with a ruler. | + | Following this formula, measure (with a ruler!) the components of $x$ along $v_1$ and $v_2$. Multiply those lengths by the scaling factors $\sigma_1$ and $\sigma_2$. Then measure out the scaled lengths along $u_1$ and $u_2$, and add those components together to get the position of the vector $y=Ax$. You can do this measuring in the units of your ruler without worrying about coordinating those units with the unit length on your plot. |
| (f) Now compute $y=Ax$ numerically and plot it on the $u_1, u_2$ plot. Does it match what you came up with in (e)? | (f) Now compute $y=Ax$ numerically and plot it on the $u_1, u_2$ plot. Does it match what you came up with in (e)? | ||
gibson/teaching/fall-2016/iam961/hw2.1474553017.txt.gz · Last modified: 2016/09/22 07:03 by gibson
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