### 0.1 Matrix Norms


**Definition 0.1.1 **  The **matrix norm** denoted $||A||$ of a matrix $A\in\mathbb{C}^{m\times n}$ induced by a vector norm $||\cdot||$ is the real number $||A|| = \max_{x\in \mathbb{C}^m/\{0\}}\frac{||Ax||}{||x||}=\max_{||x||=1}||Ax||$

**Definition 0.1.2 **  The **matrix p-q-norm** denoted $||A||_{p,q}$ of a matrix $A\in\mathbb{C}^{m\times n}$ for $1\leq p,q \leq \infty$ is the real number $||A||_{p,q} = \max_{x\in \mathbb{C}^m/\{0\}}\frac{||Ax||_p}{||x||_q}=\max_{||x||_q=1}||Ax||_p$

**Proposition 0.1.3 **  For two matrices $A\in\mathbb{C}^{m\times k}$ and $B\in\mathbb{C}^{k\times n}$, the following inequality holds for any $1\leq p,q,r\leq \infty$.
\[||AB||_{p,r}\leq ||A||_{p,q}||B||_{q,r}\]

**Proposition 0.1.4 **  The matrix $1$-norm is the max of the column sums. For any matrix $A\in\mathbb{C}^{m\times n}$ with column vectors $\{\mathbf{c}_1,\dots,\mathbf{c}_n\}$, $||A||_1 = \max_{j\in \{1,\dots,n\}}||\mathbf{c}_j||_1$.

**Proposition 0.1.5 **  Matrix multiplication by unitary matrices preserves $2$-norms. For any $A\in\mathbb{C}^{m\times n}$ and unitary $U\in\mathbb{C}^{m\times m}$, $V\in\mathbb{C}^{n\times n}$,
\[||UA||_2 = ||AV||_2 = ||A||_2\]

**Definition 0.1.6 **  The **Frobenius norm** denoted $||A||_F$ of a matrix $A\in\mathbb{C}^{m\times n}$ is the real number $||A||_F = \sqrt{\text{Tr}(A^*A)} = \sqrt{\sum_{i=1}^m{\sum_{j=1}^n{|A_{i,j}|^2}}}$.

**Proposition 0.1.7 **  Matrix multiplication by unitary matrices preserves Frobenius norms. For any $A\in\mathbb{C}^{m\times n}$ and unitary $U\in\mathbb{C}^{m\times m}$, $V\in\mathbb{C}^{n\times n}$,
\[||UA||_F = ||AV||_F = ||A||_F\]

**Corollary 0.1.8 **  For $A\in\mathbb{C}^{m\times n}$ with singular values $\sigma_1,\dots,\sigma_r$, $||A||_F = \sqrt{\sigma_1^2+\dots+\sigma_r^2}$

**0.1.9 **