Eigenvalues in a Nutshell

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Although eigenvalues are one of the most important concepts in linear algebra, some of us eigen-struggle with them without understanding their usefulness and beauty. In this talk I'll briefly review the definition of eigenvalues emphasizing the associated geometric idea and I'll show how can they be used in some applications.From the Un-Distinguished Lecture Series (http://ws.cs.ubc.ca/~udls/). The talk was given Mar. 16, 2007

Transcript of Eigenvalues in a Nutshell

  • 1. Eigenvalues in a nutshell Eigenvalues in a nutshell Mariquita Flores Garrido UDLS, March 16th 2007
  • 2. Just in case Scalar multiple of a vector x x x x x x x x 0 1 1 1 0 1 Addition of vectors v1 v1 + v2 v2
  • 3. Linear Transformations Ax = b Transformation of x by A. Rectangular matrices A R mn f : R n a R m A x = Ax mxn mx1 nx1 V. gr. 1 4 5 1 2 5 = 7 1 3 6 9
  • 4. Linear Transformations Square Matrices A R nn f : R n a R n (*endomorphism) *Stretch/Compression *Rotation *Reflection 2 0 cos sin 0 1 0 2 sin 1 0 cos
  • 5. Bonnus: Shear *Shear in x-direction *Shear in y-direction 1 k 1 0 0 1 k 1 V.gr. Shear in x-direction y x y x + ky y y x x
  • 6. Basis for a Subspace A basis in Rn is a set of n linearly independent vectors. 1 2e3 1 2 e3 e2 1 1 0 0 e1 1 = 1 0 + 1 1 + 2 0 2 0 0 1
  • 7. Basis for a Subspace Any set of n linearly independent vectors can be a basis V2 Using canonical basis: a1 a 2 e2 a1 2 V1 = a 1 e1 2 V2 Using V1, V2 ? V1 a1 = ?? a 2
  • 8. EIGENVALUES quot;Eigenquot; - quot;ownquot;, quot;peculiar toquot;, quot;characteristicquot; or quot;individual; quot;proper value. An invariant subspace under an endomorphism. If A is n x n matrix, x 0 is called an eigenvector of A if Ax = x and is called an eigenvalue of A.
  • 9. Quiz 1 Square Matrices (endomorphism) *Stretch/Compression *Rotation *Reflection 2 0 cos sin 0 1 0 2 sin 1 0 cos
  • 10. Eigen slang Characteristic polynomial: A degree n polynomial in : det(I - A) = 0 Scalars satisfying the eqn, are the eigenvalues of A. V.gr. 1 2 1 2 3 4 = 2 5 2 = 0 3 4 Spectrum (of A) : { 1, 2 , , n} Algebraic multiplicity (of i): number of roots equal to i. Eigenspace (of i): Eigenvectors never come alone! Ax = x k Ax = k x A(kx) = (kx) Geometric multiplicity (of i): number of lin. independent eigenvectors associated with i.
  • 11. Eigen slang Eigen something: Something that doesnt change under some transformation. d [e x ] = ex dx
  • 12. FAQ (yeah, sure) How old are the eigenvalues? They arose before matrix theory, in the context of differential equations. Bernoulli, Euler, 18th Century. Hilbert, 20th century. Do all matrices have eigenvalues? Yes. Every n x n matrix has n eigenvalues.
  • 13. Why are the eigenvalues important? - Physical meaning (v.gr. string, molecular orbitals ). - There are other concepts relying on eigenvalues (v.gr. singular values, condition number). - They tell almost everything about a matrix.
  • 14. Properties of a matrix reflected in its eigenvalues: 1. A singular = 0. 2. A and AT have the same s. 3. A symmetric Real s.. 4. A skew-symmetric Imaginary s.. 5. A symmetric positive definite s > 0 6. A full rank Eigenvectors form a basis for Rn. 7. A symmetric Eigenvectors can be chosen orthonormal. 8. A real Eigenvalues and eigenvectors come in conjugate pairs. 9. A symmetric Number of positive eigenvalues equals the number of positive pivots. A diagonal i = aii
  • 15. Properties of a matrix reflected in its eigenvalues: 10. A and M-1AM have the same s. 11. A orthogonal all | | = 1 12. A projector = 1,0 13. A Markov max = 1 14. A reflection = -1,1,,1 15. A rank one = vTu 16. A-1 1/(A) 17. A + cI (A) + c 18. A diagonal i = aii 19. Eigenvectors of AAT Basis for Col(A) 20. Eigenvectors of ATA Basis for Row(A) M
  • 16. Whats the worst thing about eigenvalues? Find them is painful; they are roots of the characteristic polynomial. * How long does it take to calculate the determinant of a 25 x 25 matrix? * How do we find roots of polynomials?
  • 17. WARNING: The followi