Difference between revisions of "Characteristic polynomial"

Latest revision as of 23:56, February 22, 2023

The characteristic polynomial of a square matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is given by,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\lambda) = \det(A-\lambda I)}

where $I$ is the identity matrix. The roots of this polynomial are the eigenvalues of the matrix.

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 $A={\begin{pmatrix}1&9\\4&1\end{pmatrix}}$

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_A(\lambda) = \begin{vmatrix}1-\lambda & 9 \\ 4 & 1-\lambda\end{vmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =(1-\lambda)^2-36\,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =(-5-\lambda)(7-\lambda)\,} .

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A' = \begin{pmatrix}1 & 9 \\ -4 & 1\end{pmatrix}} ,

you find the characteristic polynomial

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{A'}(\lambda) = x^2 - 2x +37\,} .

This polynomial has no roots in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} , so if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A'\,} describes a linear map between to two dimensional real vector spaces, then this map has no eigenvalue. However, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A'\,} is seen as a mapping of complex vector spaces, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{A'}(\lambda) = x^2 - 2x +37} can be factorized:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{A'}(\lambda) = (1+ 6i -\lambda)(1-6i - \lambda)\,} .

For complex spaces, the sum of algebraic multiplicities of the eigenvalues equals the degree of the characteristic polynomial.

@@ Line 1: / Line 1: @@
-{{move|Characteristic polynomial}}
 The '''characteristic polynomial''' of a square [[matrix]] <math>A</math> is given by,
 :<math>f(\lambda) = \det(A-\lambda I)</math>
-where <math>I</math> is the [[identity matrix]]. The zeros of this [[polynomial]] are the [[eigenvalue]]s of the matrix.
+where <math>I</math> is the [[identity matrix]]. The [[root (Mathematics)|root]]s of this [[polynomial]] are the [[eigenvalue]]s of the matrix.
+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]]
+[[Category:Linear Algebra]]