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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 provided by the coefficient matrix of the ordinary differential equation in the previous section:

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

The statement

`lambda = ``eig`

(A)

produces a column vector containing the eigenvalues. 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`

.

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.

The MATLAB^{®} 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
a basis with much better numerical properties than a set of eigenvectors.
The Schur decomposition of this defective example is

`[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 1.0000

The double eigenvalue is contained in the lower 2-by-2 block
of `S`

.

If `A`

is complex, `schur`

returns
the complex Schur form, which is upper triangular with the eigenvalues
of `A`

on the diagonal.