Matrix: Difference between revisions
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<math>\overrightarrow v_n = \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== | ==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\!}