## Subsection1.3.2What is a matrix norm?

A matrix norm extends the notions of an absolute value and vector norm to matrices:

###### Definition1.3.2.1. Matrix norm.

Let $\nu: \mathbb C^{m \times n} \rightarrow \mathbb R \text{.}$ Then $\nu$ is a (matrix) norm if for all $A, B \in \mathbb C^{m \times n}$ and all $\alpha \in \mathbb C$

• $A \neq 0 \Rightarrow \nu( A ) > 0$ ($\nu$ is positive definite),
• $\nu( \alpha A ) = \vert \alpha \vert \nu( A )$ ($\nu$ is homogeneous), and
• $\nu( A + B ) \leq \nu( A ) + \nu( B )$ ($\nu$ obeys the triangle inequality).

###### Homework1.3.2.1.

Let $\nu: \mathbb C^{m \times n} \rightarrow \mathbb R$ be a matrix norm.

ALWAYS/SOMETIMES/NEVER: $\nu( 0 ) = 0 \text{.}$

Hint

Review the proof on Homework 1.2.2.1.

Let $A \in \mathbb C^{m \times n} \text{.}$ Then
As we do with vector norms, we will typically use $\| \cdot \|$ instead of $\nu( \cdot )$ for a function that is a matrix norm.