Matrix: Difference between revisions
MatthewHoza (talk | contribs) |
MatthewHoza (talk | contribs) |
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Solving this polynomial we find that the eigenvalues of <math>A</math> are | Solving this polynomial we find that the eigenvalues of <math>A</math> are | ||
<math>\lambda = 3, -7</math> | <math>\lambda = 3, -7\!</math> |
Revision as of 00:11, 13 February 2009
Basics
Identity Matrix
The identity matrix, , is defined as the matrix that satisfies the condition
For any m-by-n matrix .
For example the identity matrix in R 3
Vectors
A three diemensional vector
has the matrix representation
Or more generally, an n-diemensional vector has the matrix form
Determinants
The determinant of a 2-by-2 matrix
is
Eigenvalue Analysis
Let
We must find all scalars such that the matrix equation
so we subtract by 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 I}
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 - \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} }
So 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 the solutions of the equation
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(A - \lambda I) = \begin{vmatrix} 2-\lambda & 3 \\ 3 & -6-\lambda \end{vmatrix} = 0 }
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\!}