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Vector Norms

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.

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Vector Norms

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  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 2.9208.

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

  7. Holder Inequalities Example:

  8. Bounding Norm of Product BOUND: Example:

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

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

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