# Science:Math Exam Resources/Courses/MATH307/December 2012/Question 06 (a)

MATH307 December 2012

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### Question 06 (a)

Suppose that A is a matrix with singular value decomposition ${\displaystyle A=U\Sigma V^{*}}$ where

{\displaystyle {\begin{aligned}U=&{\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}&0\\1/{\sqrt {2}}&-1/{\sqrt {2}}&0\\0&0&1\end{bmatrix}}\\\Sigma =&{\begin{bmatrix}2&0&0\\0&1&0\\0&0&1/2\end{bmatrix}}\\V=&{\begin{bmatrix}1/{\sqrt {3}}&0&2/{\sqrt {6}}\\1/{\sqrt {3}}&1/{\sqrt {2}}&-1/{\sqrt {6}}\\1/{\sqrt {3}}&-1/{\sqrt {2}}&-1/{\sqrt {6}}\end{bmatrix}}\end{aligned}}}

(a) Check all circles that apply.

${\displaystyle U}$ and ${\displaystyle V}$ are: hermitian ${\displaystyle \bigcirc }$, real symmetric ${\displaystyle \bigcirc }$, unitary ${\displaystyle \bigcirc }$, orthogonal ${\displaystyle \bigcirc }$

${\displaystyle \Sigma }$ is: hermitian ${\displaystyle \bigcirc }$, real symmetric ${\displaystyle \bigcirc }$, unitary ${\displaystyle \bigcirc }$, orthogonal ${\displaystyle \bigcirc }$

${\displaystyle A^{*}A}$ and ${\displaystyle AA^{*}}$ are: hermitian ${\displaystyle \bigcirc }$, real symmetric ${\displaystyle \bigcirc }$, unitary ${\displaystyle \bigcirc }$, orthogonal ${\displaystyle \bigcirc }$

${\displaystyle A}$ is: hermitian ${\displaystyle \bigcirc }$, real symmetric ${\displaystyle \bigcirc }$, unitary ${\displaystyle \bigcirc }$, orthogonal ${\displaystyle \bigcirc }$

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