Problem 1
State the Spectral Theorem for \(n\times n\) symmetric matrices with entries in \(\mathbb{R}\). Be sure to define all relevant terms. (6 points)
Solution. Spectral Theorem. Let \(A\) be an \(n\times n\) matrix with entries in \(\mathbb{R}\). If \(A\) is symmetric, then \(A\) is orthogonally diagonalizable. That is, there exists an orthogonal matrix \(Q\) such that
where \(D\) is a diagonal matrix.
Definitions.
- \(A\) is symmetric if \(A^{\mathrm{tr}} = A\), i.e., \(a_{ij} = a_{ji}\) for all \(i, j\).
- An \(n\times n\) matrix \(Q\) over \(\mathbb{R}\) is orthogonal if its columns \(u_1, \ldots, u_n\) are mutually orthogonal vectors of length one, i.e., \(u_i \cdot u_j = 0\) for \(i \neq j\) and \(\|u_i\| = 1\) for all \(i\). Equivalently, \(Q^{\mathrm{tr}}Q = I\), i.e., \(Q^{-1} = Q^{\mathrm{tr}}\).
In this case, the diagonal entries of \(D\) are the eigenvalues of \(A\), all of which are real, and the columns of \(Q\) are corresponding eigenvectors of \(A\).
Problem 2
Given the \(3\times 3\) matrices \(A = \left(\begin{array}{c|c} a & \begin{matrix} 0 & 0 \end{matrix} \\ \hline \begin{matrix} 0 \\ 0 \end{matrix} & A_0 \end{array}\right)\) and \(B = \left(\begin{array}{c|c} b & \begin{matrix} 0 & 0 \end{matrix} \\ \hline \begin{matrix} 0 \\ 0 \end{matrix} & B_0 \end{array}\right)\), with \(A_0\) and \(B_0\) \(2\times 2\) matrices:
- (i) What is the form taken by \(AB\)?
- (ii) If \(A\) is invertible, what form is taken by \(A^{-1}\)?
Solution.
(i) Multiplying in block form,
(ii) Expanding along the first row, \(\det A = a\,\det A_0\). So \(A\) is invertible if and only if \(a \neq 0\) and \(A_0\) is invertible. In that case,