Additionally, in the case of square matrices (thus, m = n), some (but not all) matrix norms satisfy the following condition, which is related to the fact that matrices are more than just vectors:
for all matrices and in
A matrix norm that satisfies this additional property is called a sub-multiplicative norm (in some books, the terminology matrix norm is used only for those norms which are sub-multiplicative). The set of all matrices, together with such a sub-multiplicative norm, is an example of a Banach algebra.
The definition of sub-multiplicativity is sometimes extended to non-square matrices, for instance in the case of the induced p-norm, where for and holds that . Here and are the norms induced from and , respectively, and p,q ≥ 1.
There are three types of matrix norms which will be discussed below:
Suppose a vector norm on is given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm on the space of all matrices as follows:
In particular, if the p-norm for vectors (1 ≤ p ≤ ∞) is used for both spaces and ,
then the corresponding induced operator norm is:
These induced norms are different from the "entrywise"p-norms and the Schattenp-norms for matrices treated below, which are also usually denoted by
Note: We have described above the induced operator norm when the same vector norm was used in the "departure space" and the "arrival space" of the operator . This is not a necessary restriction. More generally, given a norm on , and a norm on , one can define a matrix norm on induced by these norms:
The matrix norm is sometimes called a subordinate norm. Subordinate norms are consistent with the norms that induce them, giving
Any induced operator norm is a sub-multiplicative matrix norm: ; this follows from
Moreover, any induced norm satisfies the inequality
where ρ(A) is the spectral radius of A. For symmetric or hermitianA, we have equality in (1) for the 2-norm, since in this case the 2-norm is precisely the spectral radius of A. For an arbitrary matrix, we may not have equality for any norm; a counterexample being given by , which has vanishing spectral radius. In any case, for square matrices
we have the spectral radius formula:
In the special cases of the induced matrix norms can be computed or estimated by
which is simply the maximum absolute column sum of the matrix;
which is simply the maximum absolute row sum of the matrix;
where represents the largest singular value of matrix . There is an important inequality for the case :
where is the Frobenius norm. Equality holds if and only if the matrix is a rank-one matrix or a zero matrix. This inequality can be derived from the fact that the trace of a matrix is equal to the sum of its eigenvalues.
When p = q = 2 for the norm, it is called the Frobenius norm or the Hilbert–Schmidt norm, though the latter term is used more frequently in the context of operators on (possibly infinite-dimensional) Hilbert space. This norm can be defined in various ways:
The Frobenius norm is an extension of the Euclidean norm to and comes from the Frobenius inner product on the space of all matrices.
The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. This norm is often easier to compute than induced norms and has the useful property of being invariant under rotations, that is, for any rotation matrix . This property follows from the trace definition restricted to real matrices:
where we have used the orthogonal nature of (that is, ) and the cyclic nature of the trace (). More generally the norm is invariant under a unitary transformation for complex matrices.
The Schatten p-norms arise when applying the p-norm to the vector of singular values of a matrix. If the singular values are denoted by σi, then the Schatten p-norm is defined by
These norms again share the notation with the induced and entrywise p-norms, but they are different.
All Schatten norms are sub-multiplicative. They are also unitarily invariant, which means that for all matrices and all unitary matrices and .
The most familiar cases are p = 1, 2, ∞. The case p = 2 yields the Frobenius norm, introduced before. The case p = ∞ yields the spectral norm, which is the operator norm induced by the vector 2-norm (see above). Finally, p = 1 yields the nuclear norm (also known as the trace norm, or the Ky Fan 'n'-norm), defined as
for some positive numbers r and s, for all matrices A in . In other words, all norms on are equivalent; they induce the same topology on . This is true because the vector space has the finite dimension.
Moreover, for every vector norm on , there exists a unique positive real number such that is a sub-multiplicative matrix norm for every .
A sub-multiplicative matrix norm is said to be minimal if there exists no other sub-multiplicative matrix norm satisfying .