$$ z \in \mathbb{C} \hspace{15pt} \boldsymbol{\sim} \hspace{15pt} S \in \mathscr{L}(\mathbf{V}) \\

\hspace{3pt}\bar{z} \in \mathbb{C} \hspace{15pt} \boldsymbol{\sim} \hspace{15pt} S^* \in \mathscr{L}(\mathbf{V}) \\

\hspace{26pt} z \in \R \iff z=\bar{z} \hspace{15pt} \boldsymbol{\sim} \hspace{15pt}\text{$S$ is self-adjoint} \iff S=S^* \\

\hspace{49pt} z \in \R _{\geq0} \hspace{15pt} \boldsymbol{\sim} \hspace{15pt} \text{$S$ is positive semidefinite} \\

z \in \mathbb{C}, |z|=1 \iff zz^* = 1 \hspace{15pt} \boldsymbol{\sim} \hspace{15pt} \text{$S$ is unitary } \iff SS^* = I \hspace{26pt} $$

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Positive Semidefinite Operators

An operator $T \in \mathscr{L}(\mathbf{V})$ is positive semidefinite if it is self-adjoint and the following holds,

$$ \forall v \in V, \lang Tv \space | \space v\rang \geq 0 $$

<aside> <img src="/icons/castle_yellow.svg" alt="/icons/castle_yellow.svg" width="40px" />

Characteristics of Positive Semidefinite Operators. The following are equivalent:

  1. $T$ is positive semidefinite.
  2. $T$ is self-adjoint and all eigenvalues of $T$ are nonnegative.
  3. $T$ is diagonalizable wrt an orthonormal basis $\beta$ such that the diagonal entries of $[T]_\beta$ are nonnegative.
  4. $T$ has a unique, positive semidefinite square root.
  5. There exists some $R \in \mathscr{L}(\mathbf{V})$ for which $T = R^* R$. </aside>

Positive Definite Operators

An operator $T \in \mathscr{L}(\mathbf{V})$ is positive definite if it is self-adjoint and the following holds,

$$ \forall v \in V, v \neq 0, \lang Tv \space | \space v \rang > 0 $$

<aside> <img src="/icons/castle_yellow.svg" alt="/icons/castle_yellow.svg" width="40px" />

Cholesky Factorization. Any positive definite matrix $B \in F^{n \times n}$ can be decomposed uniquely into the product of an upper-triangular matrix $R \in F^{n \times n}$ with positive eigenvalues and its conjugate transpose $R^*$.

$$ B = R^* R $$

Isometries

A linear map $S \in \mathscr{L}(\mathbf{V}, \mathbf{W})$ is an isometry if it preserves norms.

$$ ||Sv|| = ||v|| $$

<aside> <img src="/icons/castle_yellow.svg" alt="/icons/castle_yellow.svg" width="40px" />

Characteristics of Isometries. For orthonormal bases
$\beta = e_{1:n}$ and $\gamma =f_{1:m}$ of $\mathbf{V}$ and $\mathbf{W}$ respectively, the following are equivalent:

  1. $S$ is an isometry.
  2. $S^* S = I$.
  3. For all $u , v \in V$, $\lang Su \space| \space Sv \rang = \lang u \space | \space v \rang$.
  4. The list $Se_1, \dots, Se_n$ is orthonormal.
  5. The columns of $[S]_{\beta}^\gamma$ are orthonormal with respect to the Euclidean inner product. </aside>

Unitary Operators