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Eigenvalues

Eigenvalues[m]
gives a list of the eigenvalues of the square matrix m.
Eigenvalues
gives the generalized eigenvalues of m with respect to a.
Eigenvalues
gives the first k eigenvalues of m.
Eigenvalues
gives the first k generalized eigenvalues.
  • Eigenvalues finds numerical eigenvalues if m 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 m and a 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 gives the k that are largest in absolute value.
  • Eigenvalues gives the k that are smallest in absolute value.
  • The option settings Cubics->True and Quartics->True can be used to specify that explicit radicals should be generated for all cubics and quartics.
Exact eigenvalues:
Find approximate numerical eigenvalues:
Find eigenvalues starting with 20-digit precision:
Largest 5 eigenvalues:
Multiple eigenvalues are listed multiple times:
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Exact eigenvalues:
In[1]:=
Click for copyable input
Out[1]=
 
Find approximate numerical eigenvalues:
In[1]:=
Click for copyable input
Out[1]=
 
Find eigenvalues starting with 20-digit precision:
In[1]:=
Click for copyable input
Out[1]=
 
Largest 5 eigenvalues:
In[1]:=
Click for copyable input
Out[1]=
 
Multiple eigenvalues are listed multiple times:
In[1]:=
Click for copyable input
Out[1]=
The largest 5 eigenvalues:
Explicitly use the cubic formula to give the result in terms of radicals:
Smallest eigenvalue of a Hilbert matrix:
Eigenvalues of a random matrix:
Characteristic polynomial:
The general symbolic case very quickly gets very complicated:
The expression sizes increase faster than exponentially:
Here is a 20×20 Hilbert matrix:
Compute the smallest eigenvalue exactly and give its numerical value:
Compute the smallest eigenvalue with machine-number arithmetic:
The smallest eigenvalue is not significant compared to the largest:
Using sufficient precision for the numerical computation:
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