## The Singular Value Decomposition

Suppose that $T:V\to W$ is a linear map between inner product spaces. The adjoint $T^{}$ of $T$ is the linear map $T^{}:W\to V$ satisfying $$\langle w, Tv\rangle=\langle T^{*}w,v\rangle.$$

To see that $T^{}$ exists, fix $w$ and consider the linear map $\phi_w:v\mapsto \langle w, Tv\rangle$. Since $\phi_w$ belongs to $\Hom(V,F)$, there is a unique $v’\in V$ such that $\phi_w(v)=\langle v’,v\rangle$. Then $v’=T^{}w$ by definition. The adjoint has the properties:

• $(S+T)^{\ast}=S^{\ast}+T^{\ast}$ and $(aS)^{\ast}=\overline{a}S^{\ast}$
• $(ST)^{\ast}=T^{\ast}S^{\ast}$.
• $(T^{\ast})^{\ast}=T$
• $(T^{-1})^{\ast}=(T^{\ast})^{-1}$
• The identity is its own adjoint.

The spectral theorem (over $\R$) applies to self-adjoint operators $T:V\to V$ where $V$ is an inner product space. The singular value decomposition is a generalization to operators $T:V\to W$. So let $V$ and $W$ be real inner product spaces and $T:V\to W$ a linear map.

Let $Q=T^{\ast}T$. Then $Q$ is a map from $V\to V$:

• $Q$ is self-adjoint.
• $\langle Qv, v\rangle \ge 0$ for all $v\in V$. A self adjoint operator with this property is called positive.
• The null space of $Q$ is the same as the null space of $T$.
• The range (column space) of $Q$ is the same as the column space of $T^{\ast}$.
• The ranks of $T$, $^{\ast}$, and $Q$ all coincide.

Proof:

The key point is that $\langle Qv, v\rangle=\langle T^{\ast}Tv,v\rangle = \langle Tv, Tv\rangle$. So $\langle Qv,v\rangle$ is greater than or equal to zero; and if it equals zero then $Tv=0$ and conversely. This proves that $\mathrm{null}(Q)=\mathrm{\null}(T)$.

In general:

• null space of $T^{\ast}$ is orthogonal to the range of $T$.
• range of $T^{\ast}$ is orthogonal to the null space of $T$.
• null space of $T$ is orthogonal to the range of $T^{\ast}$.
• range of $T$ is orthogonal to the nullspace of $T^{ast}$.

So since the null space of $Q$ and the null space of $T$ coincide, and $Q$ is self-adjoint, we have: $$\mathrm{range}Q = (\mathrm{null} Q)^{\perp}=(\mathrm{null}(T))^{\perp}=\mathrm{range}(T^{\ast}).$$

Also $$\dim\mathrm{range}(T)=\dim\mathrm{null}(T^{\ast})^{\perp}=\dim W-\dim\mathrm{null}(T^{\ast})=\dim\mathrm{range}(T^{\ast}).$$

Definition: If $T:V\to W$ is a linear operator, then $Q=T^{\ast}T$ is diagonalizable. Let $\Lambda$ be diagonal the matrix of $Q$ in an orthonomormal basis given by the spectral theorem, with eigenvalues listed in decreasing order. The singular values of $T$ are the entries in $\Lambda^{1/2}$ – the square roots of the eigenvalues of $Q$.

Proposition: (Singular value decomposition) Let $T:V\to W$ be a linear map of inner product spaces with singular values $s_1,\ldots, s_n$. Then there are orthonormal basis $e_1,\ldots, e_n$ and $f_1,\ldots, f_n$ for $V$ and $W$ such that $$T(v)=\sum_{i=1}^{n} s_i <v,e_i>f_i$$ for all $v\in V$. In matrix terms this basically means that $T$ is “diagonal” in the bases given by the $e$’s and $f$’s (although $T$ needn’t be square).

This is usually written like this for matrices.

Theorem: (SVD) Let $A$ be an $m\times n$ matrix over $\R$. Then there are orthogonal matrices $P$ and $Q$ such that $$A=PDQ$$ where $P$ is $m\times m$, $Q$ is $n\times n$, and $D$ is $m\times n$. The “diagonal” entries of $D$ are the singular values of $A$.

This is the matrix version of the previous statement.

Proof:

There’s an orthonormal basis of $V$ so that $Qe_i=\lambda_i e_i$ where the $\lambda_i$ are the eigenvalues of the self-adjoint operator $Q$. For the nonzero eigenvectors $e_i$ for $i=1,\dots, r$, the vectors $$f_i=\frac{Te_i}{\sqrt{\lambda_i}}$$ are orthonormal. We can complete the set of $f_i$ (if needed) to a basis of $W$. Any $v\in V$ can be written $$v=\sum_{i=1}^{n} \langle v, e_i\rangle e_i.$$ The sought-after formula comes from applying $T$ to this.