Section22Some theorems about the SVD (NM, 23 Oct)ΒΆ permalink
In what follows we will assume each matrix \(A \in \Cmn\) and that \(p=\min(m,n)\text{.}\)
Recall the Frobenius norm of a matrix:
\begin{equation*} \|A\|_F := \big(\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2\big)^{1/2}. \end{equation*}Theorem22.2
The singular values of \(A\) are the square roots of the eigenvalues of \(A^\star A\text{.}\)
Theorem22.3
If \(A\) is hermitian, the singular values of \(A\) are the absolute values of the eigenvalues of \(A\text{.}\)
Theorem22.4
If \(A\in \Cmm \text{,}\) then \(|\det(A)| =\prod_{i=1}^m \sigma_i\text{.}\)
We concluded with noting that \(A\) can be written as the sum of rank 1 matrices, and idea we will return to in the next class.
Subsection22.1Exercises
Exercise22.5
- Suppose that \(D\) is a diagonal matrix; i.e., \(D=\diag(d_{11}, d_{22}, \dots, d_{mm})\text{.}\) Show that each \(d_{ii}\) is an eigenvalue of \(D\text{.}\) What are the corresponding eigenvectors?
- Suppose that \(L\) is a lower triangular matrix. Show that each \(l_{ii}\) is an eigenvalue of \(L\text{.}\)