Vector and matrix norms

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Vector and matrix norms
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Effect of pivoting

• Pivoting allows to produce more accurate solution
  
• through matrix perturbations
• Questions
• How much round off error we can expect solving
  original system of equations?
• How much round off error pivoting reduces?
• Is it possible to know in advance whether a
  system of equations if solvable?

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• A well-conditioned problem is a problem in which
  a small change in any of the elements of the
  problem causes a small change in the solution of
  the problem.
• A ill-conditioned problem is a problem in which a
  small change in any of the elements of the
  problem causes a large change in the solution of
  the problem.

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Check for ill-conditioned matrix

• Signs of ill-conditioned matrix
• A small determinant of the matrix may be a sign
  of ill-conditioning
• det(A) ? 0 means matrix has a unique solution
• det(A) 0 means matrix has infinitely many
  solutions (or equally no solution)
• Solutions of slightly changed matrices are
  drastically different.
• The measure of ill-conditioning of a matrix is
  the condition number of a matrix,

Matlab commands det(A) cond(A)
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Vector and matrix norms

• A norm is the measure of the magnitude of a
  matrix/vector (matrix size)
• Denoted
• The same symbol is used to denote both vector and
  matrix norms, the specific type should be
  inferred from the context
• Properties

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Norms

• There are infinitely many norms that can be
  constructed.
• Most common norms
• (Euclidian
  norm)
• (maximum norm)
  
• Computation of vector norms

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Computation of matrix norms
Spectral norm
Euclidian or Frobenius norm
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Condition number

• Small values
• show small sensitivity of the solutions to
  changes in b, and characterize well-conditioned
  equations.
• Large values
• show large sensitivity of the solutions to
  changes in b, and characterize ill-conditioned
  matrix
• The higher the condition number the more
  ill-conditioned matrix is
• Value of condition number depend on the norm used