Homework 1

    1. Find a singular value decomposition for the matrix \(A = \begin{bmatrix}2 & -1 \\ 2 & 2 \end{bmatrix}\).

    2. Find a unit vector \(x\) for which \(|| A x ||\) is maximized.

  2. Find a singular value decomposition of \(A = \begin{bmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{bmatrix}\).

    1. Show that if \(A\) is an \(n \times n\) symmetric matrix, then the singular values of \(A\) are the absolute values of the eigenvalues of \(A\).

    2. Give an example to show that if \(A\) is a \(2 \times 2\) matrix which is not symmetric, then the singular values of \(A\) might not equal the absolute values of the eigenvalues of \(A\).

  4. Let \(A\) be an \(m \times n\) matrix with singular values \(\sigma_1 \ge \sigma_2 \ge \ldots \ge \sigma_n \ge 0\). Let \(v_1\) be an eigenvector of \(A^T A\) with eigenvalue \({\sigma_1}^2\) . Show that \(\sigma_2\) is the maximum value of \(|| A x ||\) where \(x\) ranges over unit vectors in \(\mathbb{R}^n\) that are orthogonal to \(v_1\).

  5. Show that if \(\{v_1 , \ldots , v_n \}\) is an orthonormal basis for \(\mathbb{R}^n\) , then \(v_1 {v_1}^T + \cdots + v_n {v_n}^T = I\).

  6. Let \(A\) be an \(m \times n\) matrix, and let \(P\) be an orthogonal \(m \times m\) matrix. Show that \(P A\) has the same singular values as \(A\).