### 0.1 Schur Decomposition


**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
1. $A_k$ has the same eigenvalues of $A$.
2. $A^{k} = (Q_1\dots Q_k)(R_k\dots R_1)$.

**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.