Q4E Find the singular values of聽聽A... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q4E (page 411)

Find the singular values of $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$

Short Answer

Expert verified

The singular values of A are $蟽_{1} = \sqrt{\frac{3 + \sqrt{5}}{2}}$ and $蟽_{2} = \sqrt{\frac{3 - \sqrt{5}}{2}}$ .

Step by step solution

Given data

Given that

$A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$

Now the singular values of A are found by first computing the eigenvalues of the square matrix

role="math" localid="1660725007450" $A^{t} A = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}] [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 1 & 2 \end{matrix}]$

Then

role="math" localid="1660724998203" $({xl}_{2} - A^{t} A) = [\begin{matrix} x - 1 & - 1 \\ - 1 & x - 2 \end{matrix}]$

Find the polynomial

Let characteristic polynomial of $A^{t} A be p (x)$ .

Then

$p (x) = \det ({xl}_{2} - A^{t} A) = (x - 1) (x - 2) - 1 = x^{2} - 3 x + 2 - 1 = x^{2} - 3 x + 1$

Thus the characteristic polynomial is $x^{2} - 3 x + 1$ .

Find the eigenvalues and singular values

Now, $x^{2} - 3 x + 1 = (x - \frac{3 - \sqrt{5}}{2}) (x - \frac{3 + \sqrt{5}}{2})$ . This implies that the roots of $p (x)$ are $\frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}$ .

Hence the eigenvalues of $A^{t} A$ are $位_{1} = \frac{3 + \sqrt{5}}{2}$ and $位_{2} = \frac{3 - \sqrt{5}}{2}$ .

Thus, the singular values of A are $蟽_{1} = \sqrt{\frac{3 + \sqrt{5}}{2}}$ and $蟽_{2} = \sqrt{\frac{3 - \sqrt{5}}{2}}$ .

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91影视

Find the singular values of $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$

Short Answer

Step by step solution

Given data

Find the polynomial

Find the eigenvalues and singular values

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91影视

Find the singular values ofA=[1101]

Short Answer

Step by step solution

Given data

Find the polynomial

Find the eigenvalues and singular values

One App. One Place for Learning.

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Find the singular values of $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$