[GSLA ch-6] Eigenvalues and Eigenvectors

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Eigenvalues and Eigenvectors

6.1 Introduction

certain vectors x are in the same direction as Ax

basic equation:

​ $ Ax = \lambda x $

how to compute:

  • let $ det(A-\lambda x)= 0 $ ,find the roots == find eigenvalues, find eigenvectors in Null space

$ A^nx = \lambda^n x $

span eigenspace

Projection: $ \lambda = 1 $ or 0

Reflection: $ \lambda = 1 $ or -1

Rotation: complex eigenvalues only


product of eigenvalues == determinant == product of pivots

sum of eigenvalues == sum of diagonal entries ( not pivots ) == trace


6.2 Diagonalizing

eigenvectors in columns of S, eigenvalues in diagonal of Λ

$ Λ = S^{-1}AS $

$ A = SΛS^{-1} $


Independent x from different λ:

  • $ c_1 λ_1 x_1 + c_2 λ_2 x_2 = 0 $
  • $ c_1 λ_2 x_2 + c_2 λ_2 x_2 = 0 $

subtract

  • $ (λ_1 - λ_2)c_1x_1 = 0 $


diagonalizability:

  • enough eigenvectors (maybe same λ) so that S is invertible

$u_k = A^k u_0 = S \Lambda^k S^{-1}u_0$

  1. $ u_0 = c_1x_1 + c_2x_2 + .. +c_nx_n $ eigenvector basis

  2. multiply $\lambda_i ^ k$

  3. add up

A,B share same eigenvector matrix S if and only if AB = BA


?Heisenberg uncertainty principle

position matrix P, momentum matrix Q, $ QP-PQ = I $

knew P still could not know Q

$ |Px||Qx| \ge \frac{1}{2}|x|^2 $


6.3 Applications to Differential Equations


6.4 Symmetric Matrices

  1. real eigenvalues
  2. orthonormal eigenvectors

Spectral Theorem

(principle of axis theorem)

$ A = Q\Lambda Q^T $


Normal Matrices $ \bar A^TA = A \bar A^T $

symmetric, skewed-symmetric, orthogonal

A has n orthonormal vectors ($ A = Q\Lambda \bar Q ^T $) if and only if A is normal

Real Eigenvalues

proof:

  • $ Ax = \lambda x $ $ \bar x ^TA = \bar x ^T \bar\lambda $ ( $ A = A^T $ )( conjugate and transpose )
  • $ \bar x^T A x = \bar x^T \lambda x $ $ \bar x ^ T A x = \bar x ^ T \bar\lambda x $

left side the same

therefor $ \lambda == \bar \lambda $


Orthonormal

proof:

  • no eigenvalues repeated
  • Allow repeated eigenvalues ( ? Schur’s Theorem )


sum of rank one projection matrices

$ A = \lambda_1x_1x_1^T + \lambda_2x_2x_2^T+… $

$ = \lambda_1P_1 +\lambda_2P2+…. $


number of positive pivots == number of positive eigenvalues


$ A = LDL^T $


6.5 Positive Definite Matrices

All λ > 0


quick way to test

  1. All pivots positive

  2. n eigenvalues positive

  3. $ x^TAx $ is positive except x = 0

  4. $ A == R^TR $ (symmetric) and R has independent columns ($ x^T R^TRX >= 0 $)

  5. n upper left determinants

R can be chosen:

  • rectangular / $ (L\sqrt D)^T $ / $ Q \sqrt\Lambda Q^T $

$x^TAx $

  • (2*2) = $ ax^2 + 2bxy + cy^2 > 0 $ ( ellipse $ z = x^2/a^2 + y^2/b^2 $ )


Application

  1. tilted ellipse $ x^TAx $
  2. lined - up ellipse $ X^T\Lambda X = 1 $
  3. rotation matrix Q

axes: eigenvectors

half-length: $ 1/\sqrt\lambda $

6.6 Similar Matrices

DEF: A similar to B (A family)

  • $ B = M^{-1}AM $


Property:

  • A and B have same eigenvalues
  • x a eigenvector of A and $ M^{-1}x $ eigenvector of B


Jordan Form

  • triple eigenvalues while one eigenvector
  • J with λ in the diagonal and 1 above
  • similar to every matrices with repeated eigenvalues λ and one eigenvector
  • λ repeated only once than J == Λ
  • Jordan Block
  • make A as simple as possible while preserving essential properties

6.7 Singular Value Decomposition

SUMMARY

$ A = U \Sigma V^T $

$ \Sigma^2 = \Lambda $ (of $ A^TA $ and $ AA^T $ )

$ U = Q $ (of $ AA^T $ in $ R^m $)

$ V = Q $ (of $ A^TA $ in $ R^n $)


orthonormal basis of row space {$ v_1, v_2, … v_r $}

orthonormal basis of null space{ $ v_{r+1}, v_{r+2},…v_n $}

orthonormal basis of column space{ $ u_1, u_2,…u_r $}

orthonormal basis of left null space { $ u_{r+1}, u_{r+2}, … u_m $}


Rotation – Stretch – Rotation

em

$ Av_1 = \sigma_1u_1 $ …

so $ AV = U\Sigma $

(m n) (n n) = (m m) (m n)

$ V $ and $ U $ are orthogonal matrices

$ \Sigma $ = ( old r*r $ \Sigma $ ) + (m-r zero rows) + ( n-r zero columns )

therefore …


when A positive definite symmetric

$ A = U \Sigma V^T = Q\Lambda Q^{T} $

7.3 Diagonalization and the Pseudoinverse

change bases

$ \Lambda { w – w } = S^{-1}{std – w} A_{std} S_{ w – std } $

$ \Sigma { v – u } = U^{-1}{std – u} A{std}V{v – std} $

Polar Decomposition

orthogonal and semidefinite

rotation and stretching

$ A = U\Sigma V^T = ( UV^T ) (V \Sigma V^T) = QH $

Pseudoinverse

$ A^+ = V\Sigma^+ U^T $

r->c

c->r