A more general subordinate matrix norm can be defined by taking different vector norms in the numerator and denominator:

Some authors denote this norm by .

A useful characterization of is given in the next result. Recall that denotes the dual of the vector norm .

**Theorem 1.** *For ,*

*Proof.* We have

where the second equality follows from the definition of dual vector norm and the fact that the dual of the dual norm is the original norm.

We can now obtain a connection between the norms of and . Here, denotes .

**Theorem 2.** *If then* .

*Proof.* Using Theorem 1, we have

If we take the – and -norms to be the same -norm then we have , where (giving, in particular, and , which are easily obtained directly).

Now we give explicit formulas for the norm when or is or and the other is a general norm.

**Theorem 3.** For ,

For ,

where

and if is symmetric positive semidefinite then

*Proof.* For (3),

with equality for , where the maximum is attained for . For (4), using the Hölder inequality,

Equality is attained for an that gives equality in the Hölder inequality involving the th row of , where the maximum is attained for .

Turning to (5), we have . The unit cube , where , is a convex polyhedron, so any point within it is a convex combination of the vertices, which are the elements of . Hence implies

and then

Hence , but trivially and (5) follows.

Finally, if is symmetric positive semidefinite let . Then, using a Cholesky factorization (which exists even if is singular) and the Cauchy–Schwarz inequality,

Conversely, for we have

so . Hence , using (5).

As special cases of (3) and (4) we have

We also obtain by using Theorem 2 and (5), for ,

The -norm has recently found use in statistics (Cape, Tang, and Priebe, 2019), the motivation being that because it satisfies

the -norm can be much smaller than the -norm when and so can be a better norm to use in bounds. The – and -norms are used by Rebrova and Vershynin (2018) in bounding the -norm of a random matrix after zeroing a submatrix. They note that the -norm of a random matrix involves maximizing over infinitely many random variables, while the -norm and -norm involve only and random variables, respectively.

The () norm is not consistent, but for any vector norm , we have