Matrix Norms#
Since matrices of the same dimensionality form a vector space, it is possible to define norms on them.
Definition: Submultiplicativity
A norm \(||\cdot||: F^{n \times n} \to \mathbb{R}\) on the square matrices in \(F^{n \times n}\) is submultiplicative if
for all \(A, B \in F^{n \times n}\).
Definition: Compatibility
We say that a norm \(||\cdot||_{F^n}: F^n \to \mathbb{R}\) on the vectors in \(F^{n}\) is compatible with a norm \(||\cdot||_{F^{n\times n}}: F^{n \times n} \to \mathbb{R}\) on the matrices in \(F^{n \times n}\) if
for all \(v \in F^n\) and all \(A \in F^{n \times n}\).
Induced Norms#
Theorem: Vector Norm induces Matrix Norm
If \(||\cdot||_{F^n}: F^n \to \mathbb{R}\) is a norm on the vectors in \(F^{n}\), then \(||\cdot||_{F^{n \times n}}: F^{n \times n} \to \mathbb{R}\) defined as the supremum
for all matrices \(A \in F^{n \times n}\) is a norm on \(F^{n \times n}\).
Definition: Induced Norm
We call \(||A||_{F^{n \times n}}\) the norm induced by \(||\cdot||_{F^n}: F^n \to \mathbb{R}\) on \(F^{n \times n}\).
Example: The Norm Induced by \(l^1\)
The norm induced on the real matrices in \(\mathbb{R}^{n \times n}\) by the \(l^1\)-norm of \(\mathbb{R}^n\) is the maximum sum of absolute values entries in a single column:
Example: The Norm Induced by \(l^{\infty}\)
The norm induced on the real matrices in \(\mathbb{R}^{n \times n}\) by the maximum norm of \(\mathbb{R}^n\) is the maximum sum of absolute values of entries in a single row:
Proof
TODO
Theorem: Compatibility of Induced Norms
If \(||\cdot||_M\) is induced by \(||\cdot||_V\), then \(||\cdot||_M\) and \(||\cdot||_V\) are compatible.
Proof
TODO