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(New page: ==Vectors== A vector <math>\overrightarrow v = v_1\hat i + v_2\hat j + v_3\hat k\!</math> has the matrix representation <math>\overrightarrow v = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end...)
 
 
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==Vectors==
==Basics==


A vector
===Identity Matrix===
The identity matrix, <math>I</math>, is defined as the matrix that satisfies the condition


<math>\overrightarrow v = v_1\hat i + v_2\hat j + v_3\hat k\!</math>
<math>IA = A\!</math>
 
For any m-by-n matrix <math>A</math>.
 
For example the identity matrix in '''R''' <sup>'''3'''</sup>
 
<math>\mathbf{I}_3 =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
</math>
 
 
===Vectors===
A three diemensional vector
 
<math>\overrightarrow v_3 = v_1\hat i + v_2\hat j + v_3\hat k\!</math>


has the matrix representation
has the matrix representation


<math>\overrightarrow v = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}</math>
<math>\overrightarrow v_3 = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}</math>


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


<math>\overrightarrow v = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}</math>
<math>\overrightarrow v_n = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}</math>
 
===Addition & Subtraction===
Only matrices with the same dimensions can be added and subtracted. If we take two matrices with dimensions <math>m \times n</math>, <math>\mathbb{A}</math> and <math>\mathbb{B}</math>, then we will get a resultant <math>m \times n</math> matrix, <math>\mathbb{C}</math>, with entries
 
<math>c_{ij} = a_{ij} + b_{ij}\!</math>
 
Similarly, for subtraction
 
<math> c_{ij} = a_{ij} - b_{ij}\!</math>
 
===Determinants===
The determinant of a 2-by-2 matrix
 
<math>A =
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
</math>
 
is
 
<math>det \mathit{A} =
\begin{vmatrix}
a & b \\
c & d \\
\end{vmatrix}
= ad - bc
</math>
 
==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>

Latest revision as of 17:49, 29 April 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}}

Addition & Subtraction

Only matrices with the same dimensions can be added and subtracted. If we take two matrices with dimensions 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 m \times n} , 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 \mathbb{A}} and 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 \mathbb{B}} , then we will get a resultant 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 m \times 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 \mathbb{C}} , with entries

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 c_{ij} = a_{ij} + b_{ij}\!}

Similarly, for subtraction

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 c_{ij} = a_{ij} - b_{ij}\!}

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

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 \mathit{A} = \begin{bmatrix} 2 & 3 \\ 3 & -6 \end{bmatrix} }

We must find all scalars 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} such that the matrix 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 (A - \lambda I)\mathbf{x} = \mathbf{0}}

so we subtract 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} 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\!}

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