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Eigenvalues and eigenvectors

`[`

also returns full matrix `V`

,`D`

,`W`

]
= eig(`A`

)`W`

whose
columns are the corresponding left eigenvectors, so that ```
W'*A
= D*W'
```

.

The eigenvalue problem is to determine the solution to the equation *A**v* = *λ**v*,
where *A* is an `n`

-by-`n`

matrix, *v* is
a column vector of length `n`

, and *λ* is
a scalar. The values of *λ* that satisfy the
equation are the eigenvalues. The corresponding values of *v* that
satisfy the equation are the right eigenvectors. The left eigenvectors, *w*,
satisfy the equation *w*’*A* = *λ**w*’.

`[`

also
returns full matrix `V`

,`D`

,`W`

]
= eig(`A`

,`B`

)`W`

whose columns are the corresponding
left eigenvectors, so that `W'*A = D*W'*B`

.

The generalized eigenvalue problem is to determine the solution
to the equation *A**v* = *λ**B**v*,
where *A* and *B* are `n`

-by-`n`

matrices, *v* is
a column vector of length `n`

, and *λ* is
a scalar. The values of *λ* that satisfy the
equation are the generalized eigenvalues. The corresponding values
of *v* are the generalized right eigenvectors. The
left eigenvectors, *w*, satisfy the equation *w*’*A* = *λ**w*’*B*.

`[___] = eig(`

,
where `A`

,`balanceOption`

)`balanceOption`

is `'nobalance'`

,
disables the preliminary balancing step in the algorithm. The default for
`balanceOption`

is `'balance'`

, which
enables balancing. The `eig`

function can return any of the
output arguments in previous syntaxes.

`[___] = eig(`

,
where `A`

,`B`

,`algorithm`

)`algorithm`

is `'chol'`

, uses
the Cholesky factorization of `B`

to compute the
generalized eigenvalues. The default for `algorithm`

depends
on the properties of `A`

and `B`

,
but is generally `'qz'`

, which uses the QZ algorithm.

If `A`

is Hermitian and `B`

is
Hermitian positive definite, then the default for `algorithm`

is `'chol'`

.

`[___] = eig(___,`

returns
the eigenvalues in the form specified by `eigvalOption`

)`eigvalOption`

using
any of the input or output arguments in previous syntaxes. Specify `eigvalOption`

as `'vector'`

to
return the eigenvalues in a column vector or as `'matrix'`

to
return the eigenvalues in a diagonal matrix.

The

`eig`

function can calculate the eigenvalues of sparse matrices that are real and symmetric. To calculate the eigenvectors of a sparse matrix, or to calculate the eigenvalues of a sparse matrix that is not real and symmetric, use the`eigs`

function.