Characteristics of Adjoint-Map Product. For $T \in \mathscr{L}(\mathbf{V}, \mathbf{W})$,

$T^*T$ is a positive-semidefinite operator on $\mathbf{V}$.
$\text{null }T^* T = \text{null } T$ and $\text{range } T^T = \text{range }T^$.
$\dim \text{range } T = \dim \text{range } T^* = \dim \text{range } T^* T$ </aside>

Singular Values

The singular values of a linear map $T \in \mathscr{L}(\mathbf{V}, \mathbf{W})$ are the principal square roots of the eigenvalues $\lambda$ of $T^*T$ listed in decreasing order and included as many times as the dimension of $E(\lambda, T^*T)$.

Characteristics of Singular Values. For $T \in \mathscr{L}(\mathbf{V}, \mathbf{W})$,

$T$ is injective $\iff$ $0$ is not a singular value of $T$.
The number of positive singular values is $\dim \text{range }T$.
$T$ is surjective $\iff$The number of positive singular values is $\dim \mathbf{W}$. </aside>

Singular Value Decomposition

The singular value decomposition of a linear map $T \in \mathscr{L}(\mathbf{V}, \mathbf{W})$ represents $T$ in terms of its singular values $s {1:m}$ and orthonormal lists $e{1:m}$ and $f_{1:m}$ in $\mathbf{V}$ and $\mathbf{W}$ respectively.

$$ Tv = s_1 \lang v \space| \space e_1\rang f_1 + \dots + s_{m}\lang v\space | \space e_m\rang f_m $$

Proof

Moreover, we can write the matrix with respect the bases these lists extend to in the following way,

$$ ([T]{e{1:\dim\mathbf{V}}}^{f_{1: \dim \mathbf{W}}})_{i, j} = \begin{cases} s_k \hspace{15pt} 1\leq i=j \leq m \\ 0 \hspace{19pt} \text{otherwise }\end{cases} $$

Which follows from the definition $f_k := \frac{Te_k}{s_k}$.