Matrix Norms

Definition

One way to measure the size or magnitude of a matrix is with an underlying vector norm. Specifically, given a vector norm, we define the corresponding matrix norm of an \(m \times n\) matrix \(\mathbf{A}\) by

\[ ||\mathbf{A}|| = \text{max}_{\mathbf{x}\neq 0} \frac{||\mathbf{Ax}||}{||\mathbf{x}||} \]

This matrix norm is said to be induced by or a subordinate to the vector norm.

Intuitively, the norm of a matrix measures the maximum stretching the matrix does to any vector, as measured in the given vector norm.

The matrix norm corresponding to the vector 1-norm is

\[ ||\mathbf{A}||_{1} = \text{max}_{j} \sum ^{m}_{i =1} |a_{ij}|, \]

which is the maximum absolute column sum of the matrix \(\mathbf{A}\).

And the matrix norm corresponding to the vector \(\infty\)-norm is

\[ ||\mathbf{A}||_{\infty} = \text{max}_{i}\sum ^{n}_{j}|a_{ij}|, \]

which is the maximum absolute row sum.

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Properties

The matrix norms we have defined satisfy the following properties, for matrices \(\mathbf{A}\) and \(\mathbf{B}\).

1. \(||\mathbf{A}|| > 0\) if \(\mathbf{A \neq O}\), (\(\mathbf{O}\) is the zero matrix)
2. \(||\gamma \mathbf{A}|| = |\gamma| \cdot ||\mathbf{A}||\) for any scalar \(\gamma\)
3. \(||\mathbf{A+B}|| \leq ||\mathbf{A}|| + \mathbf{||B||}\)
4. \(||\mathbf{AB}|| \leq ||\mathbf{A}|| \cdot ||\mathbf{B}||\)
5. \(||\mathbf{Ax}|| \leq ||\mathbf{A}|| \cdot ||\mathbf{x}||\) for any vector \(\mathbf{x}\).

References

Vector Norms