Matrix: Difference between revisions

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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\!}