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Matrix Norms

Definitions

Since \(\mathbf{R}^{m \times n}\) is isomorphic to \(\mathbf{R}^{mn}\), the definition of a matrix norm should be equivalent to the definition of a vector norm. A function \(f: \ \mathbb{R}^{m \times n} \rightarrow \mathbb{R}\) is a matrix norm if the following holds:

  1. \(f\) is nonnegative: \(f(\mathbf{A}) \geq 0\), \(\forall \mathbf{A} \in \mathbb{R}^{m \times n}\).
  2. \(f\) satisfies the triangle inequality: \(f(\mathbf{A} + \mathbf{B}) \leq f(\mathbf{A}) + f(\mathbf{B})\), for all \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{m \times n}\).
  3. \(f\) is homogeneuous: \(f( t \mathbf{A}) = |t| f(\mathbf{A})\), for all \(\mathbf{A} \in \mathbb{R}^{m \times n}\) and \(t \in \mathbb{R}\).
  4. \(f\) is definite: \(f(\mathbf{A}) = 0\) if and only if \(\mathbf{A} = \mathbf{0}\).

The Frobenius norm is defined as:

\[ ||\mathbf{A}||_F = \sqrt{\sum^m_{i = 1} \sum^n_{j = 1} |A_{ij}|^2}. \]

The p-norm is defined as:

\[ ||\mathbf{A}||_p = \sup_{x \neq 0} \frac{||\mathbf{A} \mathbf{x}||_p}{||\mathbf{x}||_p}. \]

Matrix 2-Norm

Theorem. Matrix 2-Norm

If \(\mathbf{A} \in \mathbb{R}^{m \times n}\), then there exists a unit 2-norm n-vector \(\mathbf{z}\) such that:

\[ \mathbf{A}^T \mathbf{A} \mathbf{z} = \mu^2 \mathbf{z}, \]

where \(\mu = ||\mathbf{A}||_2\).

The theorem implies that \(||\mathbf{A}||^2_2\) is a zero of \(p(\lambda) = \text{det} \left( \mathbf{A}^T \mathbf{A} - \lambda \mathbf{I} \right)\). In particular:

\[ ||\mathbf{A}||_2 = \sqrt{\lambda_{max} \left( \mathbf{A}^T \mathbf{A} \right)}. \]