## 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* =
*A**x*:

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*^{tA}*x*(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(*ω**t*), 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* =
*U**S**U* ′ ,

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