Matrix: Difference between revisions
MatthewHoza (talk | contribs) (→Basics) |
MatthewHoza (talk | contribs) |
||
Line 53: | Line 53: | ||
==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} \!