Given a matrix A, the matrix commands Eigenvectors and Eigenvalues on the Matrices submenu find scalars c and nonzero vectors v for which Av = cv. If there is a floating-point number in the matrix, you get a numerical solution. Otherwise, you get an exact symbolic solution.
These scalars and vectors are sometimes called characteristic values and characteristic vectors. The eigenvalues, or characteristic values, are roots of the characteristic polynomial.
Matrices + Eigenvalues
, eigenvalues: cosα + i sinα, cosα - i sinα
This matrix has characteristic polynomial X2 -2X cosα + 1. Replacing X by the eigenvalue cosα + i sinα and applying Simplify gives
Matrices + Eigenvalues
, eigenvalues:
+
,
-
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, eigenvalues: - .37228, 5.3723
When you choose Eigenvectors from the Matrices submenu, with each eigenvector, you get the corresponding eigenvalue. These eigenvectors are grouped by eigenvalues, and the multiplicity for each eigenvalue is indicated.
Matrices + Eigenvectors
, eigenvectors:
↔ρ where ρ is a root of Z2 - 5Z - 2
In the preceding example, ρ denotes an eigenvalue, and 1 indicates
that eigenvalue's multiplicity as a root of the characteristic polynomial.
The roots of the polynomial Z2 - 5Z - 2 are the eigenvalues computed earlier:
+
and
-
.
For
ρ = ±
, the corresponding eigenvector
is
The products of the matrix with these eigenvectors and eigenvalues are as follows.
Evaluate
=
,
=
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=
,
=
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Thus, both roots give an eigenvalue–eigenvector pair.
Matrices + Eigenvectors
, eigenvectors:
,
↔2,
↔1
In the preceding example, 2 is an eigenvalue occurring with multiplicity , and 1 is an eigenvalue occurring with multiplicity 1. The defining property Av = cv is illustrated by the following.
Evaluate
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=
, 2
=
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=
, 2
=
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