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Subsection 1.3.2 What is a matrix norm?

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

Definition 1.3.2.1. 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).

Homework 1.3.2.1.

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.

Answer

ALWAYS.

Now prove it.

Solution

Let \(A \in \mathbb C^{m \times n} \text{.}\) Then

\begin{equation*} \begin{array}{l} \nu( 0 ) \\ ~~~= ~~~~\lt 0 \cdot A = 0 \gt \\ \nu( 0 \cdot A ) \\ ~~~=~~~~\lt \| \cdot \|_\nu \mbox{ is homogeneous } \gt \\ 0 \nu( A ) \\ ~~~ = ~~~~\lt \mbox{ algebra } \gt \\ 0 \end{array} \end{equation*}
Remark 1.3.2.2.

As we do with vector norms, we will typically use \(\| \cdot \| \) instead of \(\nu( \cdot ) \) for a function that is a matrix norm.

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