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Matrices

Matrices. CS485/685 Computer Vision Dr. George Bebis. Matrix Operations. Matrix addition/subtraction Matrices must be of same size. Matrix multiplication. m x n. q x p. m x p. Condition: n = q. Identity Matrix. Matrix Transpose. Symmetric Matrices. Example:. Determinants. 2 x 2.

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Matrices

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  1. Matrices CS485/685 Computer Vision Dr. George Bebis

  2. Matrix Operations • Matrix addition/subtraction • Matrices must be of same size. • Matrix multiplication m x n q x p m x p Condition: n = q

  3. Identity Matrix

  4. Matrix Transpose

  5. Symmetric Matrices Example:

  6. Determinants 2 x 2 3 x 3 n x n

  7. Determinants (cont’d) diagonal matrix:

  8. Matrix Inverse • The inverse A-1 of a matrix A has the property: AA-1=A-1A=I • A-1 exists only if • Terminology • Singular matrix:A-1does not exist • Ill-conditioned matrix: A is close to being singular

  9. Matrix Inverse (cont’d) • Properties of the inverse:

  10. Pseudo-inverse • The pseudo-inverse A+ of a matrix A (could be non-square, e.g., m x n) is given by: • It can be shown that:

  11. Matrix trace properties:

  12. Rank of matrix • Equal to the dimension of the largest square sub-matrix of A that has a non-zero determinant. Example: has rank 3

  13. Rank of matrix (cont’d) • Alternative definition: the maximum number of linearly independent columns (or rows) of A. Therefore, rank is not 4 ! Example:

  14. Rank and singular matrices

  15. Orthogonal matrices • Notation: • A is orthogonal if: Example:

  16. Orthonormal matrices • A is orthonormal if: • Note that if A is orthonormal, it easy to find its inverse: Property:

  17. Eigenvalues and Eigenvectors • The vector vis an eigenvector of matrix A and λ is an eigenvalue of A if: • Interpretation: the linear transformation implied by A cannot change the direction of the eigenvectors v, only their magnitude. (assume non-zero v)

  18. Computing λ and v • To find the eigenvalues λ of a matrix A, find the roots of the characteristic polynomial: Example:

  19. Properties • Eigenvalues and eigenvectors are only defined for square matrices (i.e., m = n) • Eigenvectors are not unique (e.g., if v is an eigenvector, so is kv) • Suppose λ1, λ2, ..., λnare the eigenvalues of A, then:

  20. Properties (cont’d) xTAx > 0 for every

  21. Matrix diagonalization • Given A, find P such thatP-1APis diagonal (i.e., P diagonalizes A) • Take P = [v1v2 . . . vn], where v1,v2 ,. . . vnare the eigenvectors of A:

  22. Matrix diagonalization (cont’d) Example:

  23. Are all n x n matrices diagonalizable? • Only if P-1 exists (i.e., A must have n linearly independent eigenvectors, that is, rank(A)=n) • If A has n distinct eigenvalues λ1, λ2, ..., λn, then the corresponding eigevectors v1,v2 ,. . . vnform a basis: (1) linearly independent (2) span Rn

  24. Diagonalization  Decomposition • Let us assume that A is diagonalizable, then:

  25. A=PDPT= Decomposition: symmetric matrices • The eigenvalues of symmetric matrices are all real. • The eigenvectors corresponding to distinct eigenvalues are orthogonal. P-1=PT

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