matrisnorm

A matrisnorm is a function that assigns a non-negative real number to a matrix, measuring its "size" or "magnitude" in some sense. It is a generalization of the vector norm to matrices. Matrisnorms are essential in various areas of mathematics, computer science, and engineering, particularly in numerical analysis and linear algebra. They are used to quantify the error in matrix approximations, analyze the stability of algorithms, and study the properties of linear transformations.

There are several different types of matrisnorms, each with its own definition and properties. Some commonly used norms include the Frobenius norm, which is analogous to the Euclidean norm for vectors, and the induced norms, such as the 1-norm, 2-norm (spectral norm), and infinity-norm. The induced norms are defined based on the corresponding vector norms and measure the maximum scaling factor a matrix applies to any non-zero vector.

A function ||.|| is a matrisnorm if it satisfies three properties: non-negativity (||A|| >= 0, and ||A|| = 0 if and only if A is the zero matrix), absolute homogeneity (||cA|| = |c| ||A|| for any scalar c), and the triangle inequality (||A + B|| <= ||A|| + ||B|| for any matrices A and B of the same dimensions). Additionally, many matrisnorms also satisfy a submultiplicative property: ||AB|| <= ||A|| ||B||, which is crucial for error analysis. The choice of norm depends on the specific application and the properties one wishes to emphasize.