Matrix: Difference between revisions

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==Eigenvalue Analysis==
==Eigenvalue Analysis==
Let
<math>\mathit{A} =
\begin{bmatrix}
2 & 3 \\
3 & -6
\end{bmatrix}
</math>
We must find all scalars <math>\lambda</math> such that the matrix equation
<math>(A - \lambda I)\mathbf{x} = \mathbf{0}</math>
so we subtract <math>A</math> by <math>\lambda I</math>
<math>A - \lambda I =
\begin{bmatrix}
2 & 3 \\
3 & -6
\end{bmatrix}
-
\begin{bmatrix}
\lambda & 0 \\
0 & \lambda
\end{bmatrix}
=
\begin{bmatrix}
2-\lambda & 3 \\
3 & -6 - \lambda
\end{bmatrix}
</math>
So the eigenvalues of <math>A</math> are the solutions of the equation
<math>det(A - \lambda I) =
\begin{vmatrix}
2-\lambda & 3 \\
3 & -6-\lambda
\end{vmatrix}
= 0
</math>
This gives us
<math>(2 - \lambda)(-6 - \lambda) - (3)(3) = \lambda^2 + 4\lambda - 21 = 0\!</math>
Solving this polynomial we find that the eigenvalues of <math>A</math> are
<math>\lambda = 3, -7</math>\!

Revision as of 00:10, 13 February 2009

Basics

Identity Matrix

The identity matrix, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} , is defined as the matrix that satisfies the condition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle IA = A\!}

For any m-by-n matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} .

For example the identity matrix in R 3

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{I}_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} }


Vectors

A three diemensional vector

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow v_3 = v_1\hat i + v_2\hat j + v_3\hat k\!}

has the matrix representation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow v_3 = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}}

Or more generally, an n-diemensional vector has the matrix form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow v_n = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}}

Determinants

The determinant of a 2-by-2 matrix

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} }

is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle det \mathit{A} = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} = ad - bc }

Eigenvalue Analysis

Let

We must find all scalars such that the matrix equation

so we subtract by

So the eigenvalues of are the solutions of the equation

This gives us

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2 - \lambda)(-6 - \lambda) - (3)(3) = \lambda^2 + 4\lambda - 21 = 0\!}

Solving this polynomial we find that the eigenvalues of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 3, -7} \!