6.8E: Exercises for Section 6.8

Exercise \(\PageIndex{1}\)

Let \(A = \begin{bmatrix} 3 & 1 \\ -1 & 3 \end{bmatrix}\). Find the Singular Value Decomposition (SVD) of \( A \).

Answer

\(
U = \begin{bmatrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{3}{\sqrt{10}} & -\frac{1}{\sqrt{10}} \end{bmatrix}, \quad
\Sigma = \begin{bmatrix} \sqrt{10} & 0 \\ 0 & \sqrt{10} \end{bmatrix}, \quad
V = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\)

Exercise \(\PageIndex{2}\)

Let \(A=\left[\begin{array}{rrr} 4 & 0 \\ 3 &-5 \end{array}\right]\). Find the Singular Value Decomposition (SVD) of \(A\).

Exercise \(\PageIndex{3}\)

Let \( A = \begin{bmatrix} 3 & 1 & 1 \\ -1 & 3 & 1 \end{bmatrix}\). Find the Singular Value Decomposition (SVD) of \( A \).

Answer

\(U = \begin{bmatrix}
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{bmatrix}, \quad
\Sigma = \begin{bmatrix}
2\sqrt{3} & 0 & 0 \\
0 & \sqrt{10} & 0
\end{bmatrix}, \quad
V^T = \begin{bmatrix}
\frac{1}{\sqrt{6}} & -\frac{2}{\sqrt{5}} & -\frac{1}{\sqrt{30}} \\
\sqrt{\frac{2}{5}} & \frac{1}{\sqrt{5}} & -\sqrt{\frac{2}{5}} \\
\frac{1}{\sqrt{6}} & 0 & \sqrt{\frac{5}{6}}
\end{bmatrix}\)

Exercise \(\PageIndex{4}\)

Let \(A=\left[\begin{array}{rrr} 2 & -2 & 1 \\ 4 & -8 &-8 \end{array}\right]\). Find the Singular Value Decomposition (SVD) of \(A\).

Exercise \(\PageIndex{5}\)

Let \(A\) be an \(n\times n\) matrix with rank n. Show the absolute value of det(A) equals the product of its singular values.