A extra basic subordinate matrix norm might be outlined by taking totally different vector norms within the numerator and denominator:
Some authors denote this norm by 
.
A helpful characterization of 
is given within the subsequent outcome. Recall that 
denotes the twin of the vector norm 
.
Theorem 1. For 
,
Proof. We’ve
the place the second equality follows from the definition of twin vector norm and the truth that the twin of the twin norm is the unique norm.
We will now get hold of a connection between the norms of 
and 
. Right here, 
denotes 
.
Theorem 2. If then 
.
Proof. Utilizing Theorem 1, we’ve
If we take the 
– and 
-norms to be the identical 
-norm then we’ve 
, the place 
(giving, specifically, 
and 
, that are simply obtained instantly).
Now we give express formulation for the 
norm when or is 
or 
and the opposite is a basic norm.
Theorem 3. For ,
For 
,
the place
and if is symmetric constructive semidefinite then
Proof. For (3),
with equality for 
, the place the utmost is attained for 
. For (4), utilizing the Hölder inequality,
Equality is attained for an 
that offers equality within the Hölder inequality involving the 
th row of , the place the utmost is attained for 
.
Turning to (5), we’ve 
. The unit dice 
, the place ![e = [1,1,dots,1]^T](https://s0.wp.com/latex.php?latex=e+%3D+%5B1%2C1%2C%5Cdots%2C1%5D%5ET&bg=ffffff&fg=222222&s=0&c=20201002)
, is a convex polyhedron, so any level inside it’s a convex mixture of the vertices, that are the weather of 
. Therefore 
implies
after which
Therefore 
, however trivially 
and (5) follows.
Lastly, if is symmetric constructive semidefinite let 
. Then, utilizing a Cholesky factorization 
(which exists even when is singular) and the Cauchy–Schwarz inequality,
Conversely, for 
we’ve
so 
. Therefore 
, utilizing (5). 
As particular instances of (3) and (4) we’ve
We additionally get hold of by utilizing Theorem 2 and (5), for ,
The 
-norm has lately discovered use in statistics (Cape, Tang, and Priebe, 2019), the motivation being that as a result of it satisfies
the -norm might be a lot smaller than the 
-norm when 
and so generally is a higher norm to make use of in bounds. The – and 
-norms are utilized by Rebrova and Vershynin (2018) in bounding the -norm of a random matrix after zeroing a submatrix. They notice that the -norm of a random 
matrix includes maximizing over infinitely many random variables, whereas the -norm and -norm contain solely 
and 
random variables, respectively.
The (
) norm shouldn’t be constant, however for any vector norm 
, we’ve
