Math 511: Linear Algebra
Singular Value Decomposition
1. Show that $A$ and $A^T$ have the same singular values. How are their singular value decompositions related? ¶
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3. Let ¶
$$
\begin{align*}
A &= \begin{bmatrix} -2 & 8 & 20 \\ 14 & 19 & 10 \\ 2 & -2 & 1 \end{bmatrix} = \begin{bmatrix} \frac{3}{5} & -\frac{4}{5} & 0 \\ \frac{4}{5} & \frac{3}{5} & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 30 & 0 & 0 \\ 0 & 15 & 0 \\ 0 & 0 & 3 \end{bmatrix}\begin{bmatrix} \frac{1}{3} & \frac{2}{3} &\frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & -\frac{2}{3} \\ \frac{2}{3} & -\frac{2}{3} & \frac{1}{3} \end{bmatrix} \\
\end{align*}
$$
Find the closest (with respect to the Frobenious norm) matrices of rank 1 and rank 2 to $A$.
4. Let ¶
$$ A = \begin{bmatrix} 2 & 5 & 4 \\ 6 & 3 & 0 \\ 6 & 3 & 0 \\ 2 & 5 & 4 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} &\ \ \frac{1}{2} &\ \ \frac{1}{2} &\ \ \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} &\ \ \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} &\ \ \frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} &\ \ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \end{bmatrix}\begin{bmatrix} 12& 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\begin{bmatrix}\ \ \frac{2}{3} &\ \ \frac{2}{3} & \frac{1}{3} \\ -\frac{2}{3} &\ \ \frac{1}{3} & \frac{2}{3} \\ \ \ \frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \end{bmatrix} $$(a) Use the singular value decomposition of $A$ to give an orthonormal bases of $C(A^T)$ and $N(A)$.
(b) Use the singular value decomposition of $A$ to give an orthonormal bases of $C(A)$ and $N(A^T)$.
(c) Use the singular value decomposition of $A$ to determine the rank of $A$.
5. ¶
Prove that if $A\in\mathbb{R}^{n\times n}$ is a symmetric matrix with eigenvalues $\lambda_1$, $\lambda_2$, $\dots$, $\lambda_n$, then the singular value of $A$ are $|\lambda_1|$, $|\lambda_2|$, $\dots$, $|\lambda_n|$.
6. ¶
Let $A\in\mathbb{R}^{n\times n}$ be a nonsingular matrix and let $\lambda$ be an eigenvalue of $A$.
(a) Show that $\lambda$ is not equal to zero.
(b) Show that $\dfrac{1}{\lambda}$ is and eigenvalue of $A^{-1}$.
7. ¶
Show that if matrix $A$ is of the form
$$ A = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 1 \\ 0 & 0 & a \end{bmatrix} $$then $A$ must be defective.
8. ¶
Let $A$ be a matrix whose singular value decomposition is given by
$$ A = \begin{bmatrix} \frac{2}{5} & -\frac{2}{5} & -\frac{2}{5} & -\frac{2}{5} &\ \ \frac{3}{5} \\ \frac{2}{5} & -\frac{2}{5} & -\frac{2}{5} &\ \ \frac{3}{5} & -\frac{2}{5} \\ \frac{2}{5} & -\frac{2}{5} &\ \ \frac{3}{5} & -\frac{2}{5} & -\frac{2}{5} \\ \frac{2}{5} &\ \ \frac{3}{5} & -\frac{2}{5} & -\frac{2}{5} & -\frac{2}{5} \\ \frac{3}{5} &\ \ \frac{2}{5} &\ \ \frac{2}{5} &\ \ \frac{2}{5} &\ \ \frac{2}{5} \end{bmatrix}\begin{bmatrix} 100 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix}\ \ \frac{1}{2} &\ \ \frac{1}{2} &\ \ \frac{1}{2} &\ \frac{1}{2} \\ \ \ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} &\ \frac{1}{2} \\ -\frac{1}{2} & -\frac{1}{2} &\ \ \frac{1}{2} &\ \ \frac{1}{2} \\ -\frac{1}{2} &\ \ \frac{1}{2} & -\frac{1}{2} &\ \frac{1}{2} \end{bmatrix} $$(a) Determine the rank of $A$.
(b) Find an orthonormal basis for $C(A)$.
(c) Find an orthonormal basis for $N(A)$.
(d) Find an orthonormal basis for $C(A^T)$.
(e) Find an orthonormal basis for $N(A^T)$.
(f) Find the closes rank 1 matrix $B$ to $A$ with respect to the Frobenius norm.
(g) Use the singular value decomposition of $A$ to compute the distance between $A$ and $B$ with respect to the Frobenius norm.
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