Eigenvalues

Eigenvalue Decomposition

An eigenvalue and eigenvector of a square matrix A are, respectively, a scalar λ and a nonzero vector υ that satisfy

Aυ = λυ.

With the eigenvalues on the diagonal of a diagonal matrix Λ and the corresponding eigenvectors forming the columns of a matrix V, you have

AV = VΛ.

If V is nonsingular, this becomes the eigenvalue decomposition

A = VΛV^–1.

A good example is the coefficient matrix of the differential equation dx/dt = Ax:

A =
     0    -6    -1
     6     2   -16
    -5    20   -10

The solution to this equation is expressed in terms of the matrix exponential x(t) = e^tAx(0). The statement

lambda = eig(A)

produces a column vector containing the eigenvalues of A. For this matrix, the eigenvalues are complex:

lambda =
     -3.0710         
     -2.4645+17.6008i
     -2.4645-17.6008i

The real part of each of the eigenvalues is negative, so e^λt approaches zero as t increases. The nonzero imaginary part of two of the eigenvalues, ±ω, contributes the oscillatory component, sin(ωt), to the solution of the differential equation.

With two output arguments, eig computes the eigenvectors and stores the eigenvalues in a diagonal matrix:

[V,D] = eig(A)

V =
  -0.8326         0.2003 - 0.1394i   0.2003 + 0.1394i
  -0.3553        -0.2110 - 0.6447i  -0.2110 + 0.6447i
  -0.4248        -0.6930            -0.6930          

D =
  -3.0710                 0                 0         
        0           -2.4645+17.6008i        0         
        0                 0           -2.4645-17.6008i

The first eigenvector is real and the other two vectors are complex conjugates of each other. All three vectors are normalized to have Euclidean length, norm(v,2), equal to one.

The matrix V*D*inv(V), which can be written more succinctly as V*D/V, is within round-off error of A. And, inv(V)*A*V, or V\A*V, is within round-off error of D.

Multiple Eigenvalues

Some matrices do not have an eigenvector decomposition. These matrices are not diagonalizable. For example:

A = [ 1    -2    1 
      0     1    4 
      0     0    3 ]

For this matrix

[V,D] = eig(A)

produces

V =

    1.0000    1.0000   -0.5571
         0    0.0000    0.7428
         0         0    0.3714


D =

     1     0     0
     0     1     0
     0     0     3

There is a double eigenvalue at λ = 1. The first and second columns of V are the same. For this matrix, a full set of linearly independent eigenvectors does not exist.

Schur Decomposition

Many advanced matrix computations do not require eigenvalue decompositions. They are based, instead, on the Schur decomposition

A = USU ′ ,

where U is an orthogonal matrix and S is a block upper-triangular matrix with 1-by-1 and 2-by-2 blocks on the diagonal. The eigenvalues are revealed by the diagonal elements and blocks of S, while the columns of U provide an orthogonal basis, which has much better numerical properties than a set of eigenvectors.

For example, compare the eigenvalue and Schur decompositions of this defective matrix:

A = [ 6    12    19 
     -9   -20   -33 
      4     9    15 ];

[V,D] = eig(A)

V =

  -0.4741 + 0.0000i  -0.4082 - 0.0000i  -0.4082 + 0.0000i
   0.8127 + 0.0000i   0.8165 + 0.0000i   0.8165 + 0.0000i
  -0.3386 + 0.0000i  -0.4082 + 0.0000i  -0.4082 - 0.0000i


D =

  -1.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   1.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   1.0000 - 0.0000i

[U,S] = schur(A)

U =

   -0.4741    0.6648    0.5774
    0.8127    0.0782    0.5774
   -0.3386   -0.7430    0.5774


S =

   -1.0000   20.7846  -44.6948
         0    1.0000   -0.6096
         0    0.0000    1.0000

The matrix A is defective since it does not have a full set of linearly independent eigenvectors (the second and third columns of V are the same). Since not all columns of V are linearly independent, it has a large condition number of about ~1e8. However, schur is able to calculate three different basis vectors in U. Since U is orthogonal, cond(U) = 1.

The matrix S has the real eigenvalue as the first entry on the diagonal and the repeated eigenvalue represented by the lower right 2-by-2 block. The eigenvalues of the 2-by-2 block are also eigenvalues of A:

eig(S(2:3,2:3))

ans =

   1.0000 + 0.0000i
   1.0000 - 0.0000i