1. Vector Norms DEF: A norm is a function that satisfies
p-norms: The most important class
of vector norms
Example:

2. Vector Norms
Example:

3. Matrix Norm Induced by Vector Norm
DEF: the matrix norm of A (induced by the vector norm) is defined to be
DEF: If the matrix A is a square matrix

4. Matrix Norm Induced by Vector Norm
DEF: If the matrix A is a square matrix
Example:
The unit vector x that is amplified most by A is [0,1]^T, the amplification factor is 4.

5. Matrix Norm Induced by Vector Norm
DEF: If the matrix A is a square matrix
Example:
The unit vector x that is amplified most by A is the vector indicated by the dashed line, the amplification factor is

6. Holder Inequalities
Rem
Cauchy-Schwarz:
Holder Inequality:

7. Holder Inequalities
Example:

8. Bounding Norm of Product
Example:

9. Frobenius norm or Hilbert-Schmidt
DEF: Let A be a mxn matrix
REM:

10. Frobenius norm or Hilbert-Schmidt
BOUND:
BOUND:
Proof: