Determine the singular values of the 5 × 3 matrix [latex]A=\left[\begin{array}{lll}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 1 \\0 & 1 & 0 \\1 & 1 & 0\end{array}\right][/latex].

We have [latex]A^{t}=\left[\begin{array}{lllll}1 & 0 & 0 & 0 & 1 \\0 & 1 & 1 & 1 & 1 \\1 & 0 & 1 & 0 & 0\end{array}\right] \quad \text { so } \quad A^{t} A=\left[\begin{array}{lll}2 & 1 & 1 \\1 & 4 & 1 \\1 & 1 & 2\end{array}\right][/latex]. The characteristic polynomial of [latex] A^{t} A [/latex] is [latex] p\left(A^{t} A\right)=\lambda^{3}-8 \lambda^{2}+17 \lambda-10=(\lambda-5)(\lambda-2)(\lambda-1) [/latex], so the eigenvalues of [latex] A^{t} A \text { are } \lambda_{1}=s_{1}^{2}=5, \lambda_{2}=s_{2}^{2}=2, \text { and } \lambda_{3}=s_{3}^{2}=1 [/latex]. As a consequence, the singular values of A are [latex] s_{1}=\sqrt{5}, s_{2}=\sqrt{2}, s_{3}=1 [/latex], and in the singular value decomposition of A we have [latex]S=\left[\begin{array}{ccc}\sqrt{5} & 0 & 0 \\0 & \sqrt{2} & 0 \\0 & 0 & 1 \\0 & 0 & 0 \\0 & 0 & 0\end{array}\right][/latex].

Question 9.6.1: Determine the singular values of the 5 × 3 matrix A = [ 1 0 ...

Numerical Analysis

Determine the singular values of the 5 × 3 matrix

$A=\left[\begin{array}{lll}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 1 \\0 & 1 & 0 \\1 & 1 & 0\end{array}\right]$ .

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

We have

$A^{t}=\left[\begin{array}{lllll}1 & 0 & 0 & 0 & 1 \\0 & 1 & 1 & 1 & 1 \\1 & 0 & 1 & 0 & 0\end{array}\right] \quad \text { so } \quad A^{t} A=\left[\begin{array}{lll}2 & 1 & 1 \\1 & 4 & 1 \\1 & 1 & 2\end{array}\right]$ .

The characteristic polynomial of $A^{t} A$ is

$p\left(A^{t} A\right)=\lambda^{3}-8 \lambda^{2}+17 \lambda-10=(\lambda-5)(\lambda-2)(\lambda-1)$ ,

so the eigenvalues of $A^{t} A \text { are } \lambda_{1}=s_{1}^{2}=5, \lambda_{2}=s_{2}^{2}=2, \text { and } \lambda_{3}=s_{3}^{2}=1$ . As a consequence, the singular values of A are $s_{1}=\sqrt{5}, s_{2}=\sqrt{2}, s_{3}=1$ , and in the singular value decomposition of A we have

$S=\left[\begin{array}{ccc}\sqrt{5} & 0 & 0 \\0 & \sqrt{2} & 0 \\0 & 0 & 1 \\0 & 0 & 0 \\0 & 0 & 0\end{array}\right]$ .