Linear Algebra Done Right · Problem 7E.5

Singular values come from the eigenvalues of $T^*T$, the composition of $T$ with its adjoint. Since $T$ is a linear map on $\mathbb{C}^2$, you can write it as a matrix, compute $T^*T$, and find its eigenvalues—their square roots are your singular values. This approach works because singular values measure how much $T$ stretches vectors in orthogonal directions, independent of the eigenvectors of $T$ itself.

Singular values come from the eigenvalues of $T^*T$, the composition of $T$ with its adjoint. Since $T$ is a linear map on $\mathbb{C}^2$, you can write it as a matrix, compute $T^*T$, and find its eigenvalues—their square roots are your singular values. This approach works because singular values measure how much $T$ stretches vectors in orthogonal directions, independent of the eigenvectors of $T$ itself.