L16: Diagonalization

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L16 Diagonalization

• Required reading
• Lecture and lecture notes,
• Simon and Blume (23)
• Contents
• Basic facts about matrices
• Diagonalization (symmetric matrices)
• Definiteness of a matrix and eigenvalues

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Concavity and quasiconvexity

• f concave if
• Example fR R, fR R
  
• Concavity - cardinal property
• Upper contour set, C
• Ordinal implication -upper contour sets are
  convex
• Larger class quasiconcave functions
• D fX R quasiconcave if upper contour sets are
  convex
• Examples fR R fR R
• Quasiconcave but not concave?

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(Non)-singular matrix

• Square N N matrix A is nonsigular if
• Diagonal examples
• Alternative characterizations
• All vectors of A are linearly independent,
  rank(A) N
• fR R given by f(x)Ax is
• f (x) exists
• Matrix is singular if is not nonsingular

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Test for nonsingularity determinant, software

• Determinant det(A)A a number assigned to A
• Key property A0 iff A is singular
• How to find A
• 2 2 matrix
• 3 3 matrix
• Higher dimensions in practice use software

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Orthogonal Matrix P

• Dot product of x,y R.
• Length of x Euclidian norm
• Angle between x,y? x y
• Acute
• Obtuse
• Right
• D N N orthogonal matrix P P PI
  (Why?)
• QIs orthogonal matrix nonsingular?
• QWhat can you say about P

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Diagonalization

• Why do we like diagonal matrices? A
• Definiteness of matrix
• Shape of quadratic form
• Nonsingularity of A, inverse of A
• Linear dynamic systems
• Diagonal matrices result in uncoupled behavior
• Idea for symmetric matrices one can always
  change coordinates in a way that new variables
  are uncoupled
• New coordinates eigenvectors
• Entries in diagonalized matrix eigenvalues
• Q How to find them and how to use them

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Diagonalization Idea

• Consider diagonal matrix A
• Why this is useful?
• For any symmetric matrix A there exist
• N orthogonal eigenvectors
• N corresponding eigenvalues st

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Main theorem

• Theorem Let A be symmetric. There exists N N
  orthogonal
• matrix P and diagonal matrix
  st
• How can we find P and ?
• When aplied to axis v matrix A works as scalar
  
• Condition for eigenvalues?
• Condition for eigenvectors?

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Example

• A
• PS8 you will be asked to find eigenvalues of 3 3
• More generally, use sofware

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Definiteness of matrix and eigenvalues

• Theorem Let A be symmetric.
• (1) A is positive definite iff
• (2) A is positive semidefinite iff
• (3) A is indefinite iff

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Shape of a quadratic form

• A

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Definiteness of matrix and eigenvalues

• Theorem Let A be symmetric. A is invertible
  iff
• Remarks
• For non-symmetric matrices, the eigenvalues might
  be complex numbers and eigenvectors might not be
  orhogonal!

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