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Eigenvalue Problems. Solving linear systems A x = b is one part of numerical linear algebra, and involves manipulating the rows of a matrix. The second main part of numerical linear algebra is about transforming a matrix to leave its eigenvalues unchanged. A x =  x

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eigenvalue problems
Eigenvalue Problems
  • Solving linear systems Ax = b is one part of numerical linear algebra, and involves manipulating the rows of a matrix.
  • The second main part of numerical linear algebra is about transforming a matrix to leave its eigenvalues unchanged.

Ax = x

where  is an eigenvalue of A and non-zero x is the corresponding eigenvector.

what are eigenvalues
What are Eigenvalues?
  • Eigenvalues are important in physical, biological, and financial problems (and others) that can be represented by ordinary differential equations.
  • Eigenvalues often represent things like natural frequencies of vibration, energy levels in quantum mechanical systems, stability parameters.
what are eigenvectors
What are Eigenvectors
  • Mathematically speaking, the eigenvectors of matrix A are those vectors that when multiplied by A are parallel to themselves.
  • Finding the eigenvalues and eigenvectors is equivalent to transforming the underlying system of equations into a special set of coordinate axes in which the matrix is diagonal.
  • The eigenvalues are the entries of the diagonal matrix.
  • The eigenvectors are the new set of coordinate axes.
determinants
Determinants
  • Suppose we if eliminate the components of x from Ax=0 using Gaussian Elimination without pivoting.
  • We do the kth forward elimination step by subtracting ajk times row k from akk times row j, for j=k,k+1,…n.
  • Then at the end of forward elimination we have Ux=0, and Unn is the determinant of A, det(A).
  • For nonzero solutions to Ax=0 we must have det(A) = 0.
  • Determinants are defined only for square matrices.
determinant example
Determinant Example
  • Suppose we have a 33 matrix.
  • So Ax=0 is the same as:

a11x1+a12x2+a13x3 = 0

a21x1+a22x2+a23x3 = 0

a31x1+a32x2+a33x3 = 0

determinant example continued
Determinant Example (continued)
  • Step k=1:
    • subtract a21 times equation 1 from a11 times equation 2.
    • subtract a31 times equation 1 from a11 times equation 3.
  • So we have:

(a11a22-a21a12)x2+(a11a23-a21a13)x3 = 0

(a11a32-a31a12)x2+(a11a33-a31a13)x3 = 0

determinant example continued7
Determinant Example (continued)
  • Step k=2:
    • subtract (a11a32-a31a12) times equation 2 from (a11a22-a21a12) times equation 3.
  • So we have:

[(a11a22-a21a12)(a11a33-a31a13)- (a11a32-a31a12)(a11a23-a21a13)]x3 = 0

which becomes:

[a11(a22a33 –a23a32) – a12(a21a33-a23a31) + a13(a21a32-a22a31)]x3 = 0

and so:

det(A) = a11(a22a33 –a23a32) – a12(a21a33-a23a31) + a13(a21a32-a22a31)

definitions
Definitions
  • Minor Mij of matrix A is the determinant of the matrix obtained by removing row i and column j from A.
  • Cofactor Cij = (-1)i+jMij
  • If A is a 11 matrix then det(A)=a11.
  • In general,

where i can be any value i=1,…n.

a few important properties
A Few Important Properties
  • det(AB) = det(A)det(B)
  • If T is a triangular matrix,

det(T) = t11t22…tnn

  • det(AT) = det(A)
  • If A is singular then det(A)=0.
  • If A is invertible then det(A)0.
  • If the pivots from Gaussian Elimination are d1, d2,…,dn then

det(A) = d1d2dn

where the plus or minus sign depends on whether the number of row exchanges is even or odd.

characteristic equation
Characteristic Equation
  • Ax = x can be written as

(A-I)x = 0

which holds for x0, so (A- I) is singular and

det(A-I) = 0

  • This is called the characteristic polynomial. If A is nn the polynomial is of degree n and so A has n eigenvalues, counting multiplicities.
example
Example
  • Hence the two eigenvalues are 1 and 5.
example continued
Example (continued)
  • Once we have the eigenvalues, the eigenvectors can be obtained by substituting back into (A-I)x = 0.
  • This gives eigenvectors (1 -1)T and (1 1/3)T
  • Note that we can scale the eigenvectors any way we want.
  • Determinant are not used for finding the eigenvalues of large matrices.
positive definite matrices
Positive Definite Matrices
  • A complex matrix A is positive definite if for every nonzero complex vector x the quadratic form xHAx is real and:

xHAx > 0

where xH denotes the conjugate transpose of x (i.e., change the sign of the imaginary part of each component of x and then transpose).

eigenvalues of positive definite matrices
Eigenvalues of Positive Definite Matrices
  • If A is positive definite and  and x are an eigenvalue/eigenvector pair, then:

Ax = x  xHAx = xHx

  • Since xHAx and xHx are both real and positive it follows that  is real and positive.
properties of positive definite matrices
Properties of Positive Definite Matrices
  • If A is a positive definite matrix then:
    • A is nonsingular.
    • The inverse of A is positive definite.
    • Gaussian elimination can be performed on A without pivoting.
    • The eigenvalues of A are positive.
hermitian matrices
Hermitian Matrices
  • A square matrix for which A = AH is said to be an Hermitian matrix.
  • If A is real and Hermitian it is said to be symmetric, and A = AT.
  • Every Hermitian matrix is positive definite.
  • Every eigenvalue of an Hermitian matrix is real.
  • Different eigenvectors of an Hermitian matrix are orthogonal to each other, i.e., their scalar product is zero.
eigen decomposition
Eigen Decomposition
  • Let 1,2,…,n be the eigenvalues of the nn matrix A and x1,x2,…,xn the corresponding eigenvectors.
  • Let  be the diagonal matrix with 1,2,…,n on the main diagonal.
  • Let X be the nn matrix whose jth column is xj.
  • Then AX = X, and so we have the eigen decomposition of A:

A = XX-1

  • This requires X to be invertible, thus the eigenvectors of A must be linearly independent.
powers of matrices
Powers of Matrices
  • If A = XX-1 then:

A2 = (XX-1)(XX-1) = X(X-1X)X-1 = X2X-1

Hence we have:

Ap = XpX-1

  • Thus, Ap has the same eigenvectors as A, and its eigenvalues are 1p,2p,…,np.
  • We can use these results as the basis of an iterative algorithm for finding the eigenvalues of a matrix.
the power method
The Power Method
  • Label the eigenvalues in order of decreasing absolute value so |1|>|2|>… |n|.
  • Consider the iteration formula:

yk+1 = Ayk

where we start with some initial y0, so that:

yk = Aky0

  • Then yk converges to the eigenvector x1 corresponding the eigenvalue 1.
proof
Proof
  • We know that Ak = XkX-1, so:

yk = Aky0 = XkX-1y0

  • Now we have:
  • The terms on the diagonal get smaller in absolute value as k increases, since 1 is the dominant eigenvalue.
proof continued
Proof (continued)
  • So we have
  • Since 1k c1 x1 is just a constant times x1 then we have the required result.
example22
Example
  • Let A = [2 -12; 1 -5] and y0=[1 1]’
  • y1 = -4[2.50 1.00]’
  • y2 = 10[2.80 1.00]’
  • y3 = -22[2.91 1.00]’
  • y4 = 46[2.96 1.00]’
  • y5 = -94[2.98 1.00]’
  • y6 = -190[2.99 1.00]’
  • The iteration is converging on a scalar multiple of [3 1]’, which is the correct dominant eigenvector.
rayleigh quotient
Rayleigh Quotient
  • Note that once we have the eigenvector, the corresponding eigenvalue can be obtained from the Rayleigh quotient:

dot(Ax,x)/dot(x,x)

where dot(a,b) is the scalar product of vectors a and b defined by:

dot(a,b) = a1b1+a2b2+…+anbn

  • So for our example, 1 = -2.
scaling
Scaling
  • The 1k can cause problems as it may become very large as the iteration progresses.
  • To avoid this problem we scale the iteration formula:

yk+1 = A(yk /k+1)

where k+1 is the component of Ayk with largest absolute value.

example with scaling
Example with Scaling
  • Let A = [2 -12; 1 -5] and y0=[1 1]’
  • Ay0 = [-10 -4]’ so 1=-10 and y1=[1.00 0.40]’.
  • Ay1 = [-2.8 -1.0]’ so 2=-2.8 and y2=[1.0 0.3571]’.
  • Ay2 = [-2.2857 -0.7857]’ so 3=-2.2857 and y3=[1.0 0.3437]’.
  • Ay3 = [-2.1250 -0.7187]’ so 4=-2.1250 and y4=[1.0 0.3382]’.
  • Ay4 = [-2.0588 -0.6912]’ so 5=-2.0588 and y5=[1.0 0.3357]’.
  • Ay5 = [-2.0286 -0.6786]’ so 6=-2.0286 and y6=[1.0 0.3345]’.
  •  is converging to the correct eigenvector -2.
scaling factor
Scaling Factor
  • At step k+1, the scaling factor k+1 is the component with largest absolute value is Ayk.
  • When k is sufficiently large Ayk  1yk.
  • The component with largest absolute value in 1yk is 1 (since yk was scaled in the previous step to have largest component 1).
  • Hence, k+1  1 as k  .
matlab code
MATLAB Code

function [lambda,y]=powerMethod(A,y,n)

for (i=1:n)

y = A*y;

[c j] = max(abs(y));

lambda = y(j);

y = y/lambda;

end

convergence
Convergence
  • The Power Method relies on us being able to ignore terms of the form (j/1)k when k is large enough.
  • Thus, the convergence of the Power Method depends on |2|/|1|.
  • If |2|/|1|=1 the method will not converge.
  • If |2|/|1| is close to 1 the method will converge slowly.
orthonormal vectors
Orthonormal Vectors
  • A set S of nonzero vectors are orthonormal if, for every x and y in S, we have dot(x,y)=0 (orthogonality) and for every x in S we have ||x||2=1 (length is 1).
similarity transforms
Similarity Transforms
  • If T is invertible, then the matrices A and B=T-1AT are said to be similar.
  • Similar matrices have the same eigenvalues.
  • If x is the eigenvector of A corresponding to eigenvalue , then the eigenvector B corresponding to eigenvalue  is T-1x.

Ax=x  TBT-1x=x  B(T-1x)=(T-1x)

the qr algorithm
The QR Algorithm
  • The QR algorithm for finding eigenvalues is based on the QR factorisation that represents a matrix A as:

A = QR

where Q is a matrix whose columns are orthonormal, and R is an upper triangular matrix.

  • Note that QHQ = I and Q-1=QH.
  • Q is termed a unitary matrix.
qr algorithm without shifts
QR Algorithm without Shifts

A0 = A

for k=1,2,…

QkRk = Ak

Ak+1 = RkQk

end

Since:

Ak+1 = RkQk = Qk-1AkQk

then Ak and Ak+1 are similar and so have the same eigenvalues.

Ak+1 tends to an upper triangular matrix with the same eigenvalues as A. These eigenvalues lie along the main diagonal of Ak+1.

qr algorithm with shift
QR Algorithm with Shift

Since:

Ak+1 = RkQk + sI

= Qk-1(Ak – sI)Qk +sI

= Qk-1AkQk

so once again Ak and Ak+1 are similar and so have the same eigenvalues.

A0 = A

for k=1,2,…

s = Ak(n,n)

QkRk = Ak - sI

Ak+1 = RkQk + sI

end

The shift operation subtracts s from each eigenvalue of A, and speeds up convergence.

matlab code for qr algorithm
MATLAB Code for QR Algorithm
  • Let A be an nn matrix

n = size(A,1);

I = eye(n,n);

s = A(n,n); [Q,R] = qr(A-s*I); A = R*Q+s*I

  • Use the up arrow key in MATLAB to iterate or put a loop round the last line.
deflation
Deflation
  • The eigenvalue at A(n,n) will converge first.
  • Then we set s=A(n-1,n-1) and continue the iteration until the eigenvalue at A(n-1,n-1) converges.
  • Then set s=A(n-2,n-2) and continue the iteration until the eigenvalue at A(n-2,n-2) converges, and so on.
  • This process is called deflation.