# Thread: Norm of not-square matrixes

1. How does definition of matrix norm (induced and not induced - Frobenius Norm, for example) change for not-square matrixes?

- submultiplicative property ||AB||≤||A||||B|| of a generic matrix norm;

- property ||A||≥ρ(A), where ρ(A) is the greatest eigenvalue of matrix A as absolute value;

2.

3. Originally Posted by doctor_cat
How does definition of matrix norm (induced and not induced - Frobenius Norm, for example) change for not-square matrixes?

- submultiplicative property ||AB||≤||A||||B|| of a generic matrix norm;

- property ||A||≥ρ(A), where ρ(A) is the greatest eigenvalue of matrix A as absolute value;
The norm of an operator A between normed spaces is the supremum of the norm of
A(x) where x ranges over elements in the domain of norm less than or equal to 1.

4. And what about the properties?

5. Originally Posted by doctor_cat
It is a norm, with the propertiess of a norm.

6. So aren't submultiplicative property and ||A||≥ρ(A) property fundamental for a norm?

They are valid only for square matrixes and nothing more.

7. Originally Posted by doctor_cat
So aren't submultiplicative property and ||A||≥ρ(A) property fundamental for a norm?

The submultiplicative property applies to a normed algebra, not to every normed space.

Originally Posted by doctor_cat
They are valid only for square matrixes and nothing more.
nope

You need square matrices for a normed algebra , but not for just a normed space.

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