Matrix Norms

Definition: is a matrix norm on matrices if it is a vector norm on an dimensional space:

1. , and
2. 
3. 

Definition: Let

They are called mutually consistent if ,

Example: is the ``max norm".

, is the Frobenius norm.

Definition: Given , let be a vector norm on , be a vector norm on . Then

is called an operator norm or induced norm. The geometric interpretation of such a norm is that it is the maximum length of a unit vector after transformation by . Furthermore, an operator norm is a matrix norm (i.e. satisfies the property listed above).

Definition: A real square matrix is orthogonal, if . For an orthogonal matrix, all the rows and columns have and are orthogonal to one another.

Some properties of matrix and vector norms:

1. for operator and (Frobenius norm + vector 2 norm)
2. for operator and Froberiur norm.
3. Max norm is not an operator norm.
4. if are orthogonal for Frobenius and operator norm induced by .
5. max absolute row sum.
6. max absolute column sum.

7. , where is the absolute value of the largest eigenvalue in magnitude of the matrix . (This is called the spectral radius of the matrix.) If is symmetric, corresponds to the maximum absolute eigenvalue, i.e.