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Characteristic polynomial (view source)
Revision as of 23:56, February 22, 2023
, 23:56, February 22, 2023no edit summary
The '''characteristic polynomial''' of a matrix <math>A</math> is given by,
The '''characteristic polynomial''' of a square [[matrix]] <math>A</math> is given by,
:<math>|A-\lambda I|=0</math>
:<math>f(\lambda) = \det(A-\lambda I)</math>
where <math>I</math> is the [[identity matrix]].
where <math>I</math> is the [[identity matrix]]. The [[root (Mathematics)|root]]s of this [[polynomial]] are the [[eigenvalue]]s of the matrix.
[[Category:mathematics]]
The degree of this polynomial equals the length of the matrix's side. The number of roots therefore is not greater than this number.
== Example ==
Take <math>A = \begin{pmatrix}1 & 9 \\ 4 & 1\end{pmatrix}</math>
Then <math>f_A(\lambda) = \begin{vmatrix}1-\lambda & 9 \\ 4 & 1-\lambda\end{vmatrix}</math><math>=(1-\lambda)^2-36\,</math><math>=(-5-\lambda)(7-\lambda)\,</math>.
So, the roots of the characteristic polynomial are {-5, 7} - and these are the eigenvalues of the matrix. If you look at the slightly different matrix
<math>A' = \begin{pmatrix}1 & 9 \\ -4 & 1\end{pmatrix}</math>,
you find the characteristic polynomial
<math>f_{A'}(\lambda) = x^2 - 2x +37\,</math>.
This polynomial has no roots in <math>\mathbb{R}</math>, so if <math>A'\,</math> describes a linear map between to two dimensional real vector spaces, then this map has no eigenvalue. However, if <math>A'\,</math> is seen as a mapping of complex vector spaces, <math>f_{A'}(\lambda) = x^2 - 2x +37</math> can be factorized:
<math>f_{A'}(\lambda) = (1+ 6i -\lambda)(1-6i - \lambda)\,</math>.
For complex spaces, the sum of algebraic multiplicities of the eigenvalues equals the degree of the characteristic polynomial.
[[Category:Linear Algebra]]