Science:Math Exam Resources/Courses/MATH307/April 2013/Question Section 201 07 (a)

MATH307 April 2013

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Question Section 201 07 (a)
Suppose that the matrix A has the following singular value decomposition: $A={\begin{bmatrix}0&1\\1&0\end{bmatrix}}{\begin{bmatrix}\sigma _{1}&0&0\\0&\sigma _{2}&0\end{bmatrix}}{\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}&0\\0&0&1\\-1{\sqrt {2}}&1/{\sqrt {2}}&0\end{bmatrix}}$ where $\sigma _{1}\geq \sigma _{2}\geq 0$. (a) What are $\|A\|$ and rank(A)?

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Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If we define $U={\begin{bmatrix}0&1\\1&0\end{bmatrix}},\ \ \Sigma ={\begin{bmatrix}\sigma _{1}&0&0\\0&\sigma _{2}&0\end{bmatrix}},\ \ V={\begin{bmatrix}{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\0&0&1\\-{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\end{bmatrix}}$ Matrices U and V are orthogonal matrices because their columns form a basis of orthonormal vectors, all the entries are real, and the sum of the column entries are equal to 1. Matrix $\Sigma$ is a diagonal matrix. A property of orthogonal matrices is that they have no effect on the length of a norm. $\displaystyle ||A||=||\Sigma ||=\sigma _{1}$ The norm of matrix A is equal to the largest value on the diagonal of matrix $\Sigma$. The rank of A is equal to the number of pivot columns in the diagonal matrix $\Sigma$: $\dim(A)=2$