Definition 0.1.1 A similar matrix $A\in\mathbb{C}^{n\times n}$ to another matrix $B\in\mathbb{C}^{n\times n}$ is a matrix where there exists a non-singular matrix $S\in\mathbb{C}^{n\times n}$ such that $A = S^{-1}BS$.
Definition 0.1.2 The singularity transformation for a non-singular matrix $S\in\mathbb{C}^{n\times n}$ is the mapping $\text{Sim}_S : \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$ defined by
\[\text{Sim}_S(A) = S^{-1}AS\]
Definition 0.1.3 A matrix $A\in\mathbb{C}^{n\times n}$ is diagonalizable iff there exists a non-singular $S\in\mathbb{C}^{n\times n}$ such that $S^{-1}AS$ is a diagonal matrix.
Definition 0.1.4 A matrix $A\in\mathbb{C}^{n\times n}$ is unitary diagonalizable iff there exists a unitary $U\in\mathbb{C}^{n\times n}$ such that $U^* AU$ is a diagonal matrix.
Definition 0.1.5 A matrix $A\in\mathbb{C}^{n\times n}$ is unitary triangularizable iff there exists a unitary $U\in\mathbb{C}^{n\times n}$ such that $U^* AU$ is upper triangular matrix.
Definition 0.1.6 The Schur decomposition of $A\in\mathbb{C}^{n\times n}$ is a unitary Q and upper triangular T such that
\[A = QTQ^*\]
Proposition 0.1.7 A matrix is diagonalizable iff A has n eignenvectors that form a basis of $\mathbb{C}^n$.
Proposition 0.1.8 A matrix is unitary diagonalizable iff $A^*A = AA^*$
Proposition 0.1.9 Any matrix $A\in\mathbb{C}^{n\times n}$ has a Schur decomposition $A = QTA^*$.
Proposition 0.1.10 For $A\in\mathbb{C}^{n\times n}$ let $\{A_k\}$ be the sequence of matrices defined by $A_0= A$, $A_{k-1} = Q_kR_k$, and $A_k = R_k Q_k$, then
Theorem 0.1.11 If $A\in\mathbb{C}^{n\times n}$ is a matrix with eigenvalues $\lambda_1,\dots,\lambda_n$ such that $|\lambda_1| > \dots > |\lambda_n| > 0$, then the sequence of matrices $\{T_k\}$ defined by $T_0= A$, $T_{k-1} = Q_kR_k$, and $T_k = R_k Q_k$ converges to $T$ and $Q_1Q_2\dots$ converges to $Q$ in the Schur decomposition $A = QTQ^*$.
Algorithm 0.1.12 The pure QR Schur decomposition algorithm can be used to iteratively compute the Schur decomposition of a matrix $A=QTQ^*$.
Def 0.1.13 An matrix $H\in\mathbb{C}^{n\times n}$ is upper Hessenberg iff $H_{i,j}=0$ for $i>j+1$.
Proposition 0.1.14 All square matrices are unitarily similar to an upper hessenber matrix.
Algorithm 0.1.15 The practical QR Schur decomposition algorithm can be used to iteratively compute the Schur decomposition of a matrix $A=QTQ^*$ more efficiently.