# Subsection13.1Singular Values

The singular values $\{\sigma_1, \sigma_2, \dots, \sigma_m\}$ of a matrix $A$ can be defined in several ways:

• the square roots of the eigenvalues of $B=A^\star A\text{,}$
• the lengths of the semiaxes of the hyperellipse that is the image of the unit circle under the linear transformation defined by $A\text{.}$
• The diagonal entries of the diagonal matrix $\Sigma$ when we write $A$ has a factorisation $A=U \Sigma V^\star$ where $U$ and $V$ are unitary matrices, and $\Sigma$ is a diagonal matrix. (We'll be a little more precise below).

Please read Lecture 4 of Trefethen and Bau. T+B make the important observation in their introduction: the image of the unit sphere under any $m \times n$ matrix is a hyperellipse.

Suppose that $\{u_i\}_{i=1}^m$ is a set of orthonormal vectors, such that principal semiaxes of the hyperellipse are $\{\sigma_i u_i\}\text{.}$ These scalars $\sigma_1, \sigma_2, \dots, \sigma_m$ are the singular values of $A$, and encapsulate many important properties of $A\text{.}$

# Subsection13.2Singular Value Decomposition

The rest of the exposition followed Lecture 4 of Trefethen and Bau pretty closely. In summary, the SVD of $A \in \Cmn$ is

\begin{equation*} AV = \Sigma U, \end{equation*}

where

• $U=(u_1 | u_2 | u_3 | \dots | u_m) \text{,}$ the matrix that has as its columns the orthoronormal vectors in the direction of the principal semiaxes of the image of the unit sphere $S \in \Rn \text{.}$
• $\Sigma = \diag(\sigma_1, \sigma_2, \dots, \sigma_n)$ is the diagonal matrix containing the lengths of the principal semiaxes. We order them as \begin{equation*} \sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_n \geq 0. \end{equation*}
• Then $V=(v_1 | v_2 | v_3 | \dots | v_n)$ is the matrix whose columns are the preimages of the principal semiaxes, i.e., $A v_i = \sigma_i u_i.$ It should be clear that the $\{v_i\}$ form an orthonormal set, so $V^{-1}=V^\star.$

# Subsection13.3Exercises

##### Exercise13.1

Prove that the product of two unitary matrices in $\Cmm$ is unitary.

##### Exercise13.2

Write down the SVDs of the following matrices.

1. $\begin{pmatrix} 4 \amp 0 \\ 0 \amp -2 \end{pmatrix}$
2. $\begin{pmatrix} 0 \amp 4 \\ -2 \amp 0 \end{pmatrix}$
3. $\begin{pmatrix} 4 \amp -2 \\ 0 \amp 0 \end{pmatrix}$
##### Exercise13.3

The matrices $A$ and $B$ in $\Cmm$are unitarily equivalent if there exists a untiary matrix, $Q \in \Cmm$ such that $A=QBQ^\star\text{.}$

1. Show that, if $A$ and $B$ are unitarily equivalent, then they have the same singular values.
2. Suppose that $A, B \in \Cmm$ have the same singular values. Must they be unitarily equivalent?