3. Norms
A norm is how we measure the length/size of vectors and matrices.
An example is relative error:
, . . Sub-additivity / triangle inequality.
Common Vector Norms
The infinity norm is the maximum value of a vector
This is very punishing of outliers.
The 1 norm is the sum of the absolute values
The 2 norm is the square root of sum the squares.
These are all examples of
The limit is what happens when
Isometries
Isometries are operations that do not change the norms.
- Permutations
.
- Orthogonal matrices
.
Norm Equivalence
Given two norms
Example. The infinity norm is less than the 1 norm less than
Note that they are equivalent when
Matrix Norms
Suppose
We can also calculate the induced norm, or operator norm.
A nice property of induced norms is that they are subordinate and submultiplicative. See Cauchy-Schwarz. 1.4. Geometry of the real numbers.
This is not a property of all norms. For example, suppose we want ot take the max norm
Matrix Isometries
Suppose
This property also holds for frobenius norms.
Matrix 2-norm (spectral norm)
Suppose we have