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

Vector Norms Example:

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

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.

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 2.9208.

Holder Inequalities Rem Cauchy-Schwarz: Holder Inequality:

Holder Inequalities Example:

Bounding Norm of Product Example:

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

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