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DOCUMENTATION CENTER SEARCH
Mathematica
>
Mathematics and Algorithms
>
Matrices and Linear Algebra
>
Matrix Operations
>
Built-in
Mathematica
Symbol
Vectors and Matrices
Eigenvalues and Eigenvectors
Tutorials »
|
Eigenvectors
Eigensystem
SingularValueList
CharacteristicPolynomial
Det
Tr
PositiveDefiniteMatrixQ
See Also »
|
Matrices and Linear Algebra
Matrix Operations
More About »
Eigenvalues
Eigenvalues
[
m
]
gives a list of the eigenvalues of the square matrix
.
Eigenvalues
[{
m
,
a
}]
gives the generalized eigenvalues of
with respect to
.
Eigenvalues
[
m
,
k
]
gives the first
eigenvalues of
.
Eigenvalues
[{
m
,
a
},
k
]
gives the first
generalized eigenvalues.
MORE INFORMATION
Eigenvalues
finds numerical eigenvalues if
contains approximate real or complex numbers.
Repeated eigenvalues appear with their appropriate multiplicity.
An
×
matrix gives a list of exactly
eigenvalues, not necessarily distinct.
If they are numeric, eigenvalues are sorted in order of decreasing absolute value.
The eigenvalues of a matrix
are those
for which
for some non-zero eigenvector
.
The generalized eigenvalues of
with respect to
are those
for which
.
When matrices
and
have a dimension-
shared null space, then
of their generalized eigenvalues will be
Indeterminate
.
Ordinary eigenvalues are always finite; generalized eigenvalues can be infinite.
For numeric eigenvalues,
Eigenvalues
[
m
,
k
]
gives the
that are largest in absolute value.
Eigenvalues
[
m
, -
k
]
gives the
that are smallest in absolute value.
Eigenvalues
[
m
,
spec
]
is always equivalent to
Take
[Eigenvalues[
m
],
spec
]
.
The option settings
Cubics
->
True
and
Quartics
->
True
can be used to specify that explicit radicals should be generated for all cubics and quartics.
SparseArray
objects can be used in
Eigenvalues
.
EXAMPLES
CLOSE ALL
Basic Examples
(6)
In[1]:=
Out[1]=
In[2]:=
Out[2]=
Exact eigenvalues:
In[1]:=
Out[1]=
Find approximate numerical eigenvalues:
In[1]:=
Out[1]=
Find eigenvalues starting with 20-digit precision:
In[1]:=
Out[1]=
Largest 5 eigenvalues:
In[1]:=
Out[1]=
Multiple eigenvalues are listed multiple times:
In[1]:=
Out[1]=
Generalizations & Extensions
(1)
Options
(1)
Applications
(2)
Properties & Relations
(1)
Possible Issues
(2)
Neat Examples
(1)
SEE ALSO
Eigenvectors
Eigensystem
SingularValueList
CharacteristicPolynomial
Det
Tr
PositiveDefiniteMatrixQ
TUTORIALS
Vectors and Matrices
Eigenvalues and Eigenvectors
MORE ABOUT
Matrices and Linear Algebra
Matrix Operations
RELATED LINKS
Demonstrations with Eigenvalues
(
Wolfram Demonstrations Project
)
NKS|Online
(
A New Kind of Science
)
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