Eigenvalue: Difference between revisions
imported>Michael Underwood No edit summary |
imported>Michael Underwood No edit summary |
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That is, to find a number <math>\lambda</math> and a vector <math>\vec{v}</math> that together satisfy | That is, to find a number <math>\lambda</math> and a vector <math>\vec{v}</math> that together satisfy | ||
:<math>A\vec{v}=\lambda\vec{v}\ .</math> | :<math>A\vec{v}=\lambda\vec{v}\ .</math> | ||
What this equation says is that even though <math>A</math> is a matrix its action on <math>\vec{v}</math> is the same as multiplying it by the number <math>\lambda</math>. | What this equation says is that even though <math>A</math> is a matrix its action on <math>\vec{v}</math> is the same as multiplying it by the number <math>\lambda</math>. | ||
This means that the vector <math>\vec{v}</math> and the vector <math>A\vec{v}</math> are [[parallel]] (or [[anti-parallel]] if <math>\lambda</math> is negative). | |||
Note that generally this will ''not'' be true. This is most easily seen with a quick example. Suppose | |||
:<math>A=\begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}</math> and <math>\vec{v}=\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}\ .</math> | :<math>A=\begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}</math> and <math>\vec{v}=\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}\ .</math> | ||
Then their [[matrix multiplication|matrix product]] is | Then their [[matrix multiplication|matrix product]] is | ||
:<math>A\vec{v}=\begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}\begin{pmatrix} v_1 \\ v_2 \end{pmatrix} | :<math>A\vec{v}=\begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}\begin{pmatrix} v_1 \\ v_2 \end{pmatrix} | ||
=\begin{pmatrix}a_{11}v_1+a_{12}v_2 \\ a_{21}v_1+a_{22}v_2 \end{pmatrix}</math> | =\begin{pmatrix}a_{11}v_1+a_{12}v_2 \\ a_{21}v_1+a_{22}v_2 \end{pmatrix}</math> | ||
whereas | whereas the [[scalar]] product is | ||
:<math>\lambda\vec{v}=\begin{pmatrix} \lambda v_1 \\ \lambda v_2 \end{pmatrix}\ .</math> | :<math>\lambda\vec{v}=\begin{pmatrix} \lambda v_1 \\ \lambda v_2 \end{pmatrix}\ .</math> | ||
Obviously then <math>A\vec{v}\neq \lambda\vec{v}</math> unless | |||
<math>\lambda v_1 = a_{11}v_1+a_{12}v_2</math> and [[simultaneous equations|simultaneously]] <math>\lambda v_2 = a_{21}v_1+a_{22}v_2</math>, | |||
and it is easy to pick numbers for the entries of <math>A</math> and <math>\vec{v}</math> such that this cannot happen for any value of <math>\lambda</math>. | |||
==The eigenvalue equation== | |||
So where did the eigenvalue equation <math>\text{det}(A-\lambda I)=0</math> come from? Well, we assume that we know the matrix <math>A</math> and want to find a number <math>\lambda</math> and a non-zero vector <math>\vec{v}</math> so that <math>A\vec{v}=\lambda\vec{v}</math>. (Note that if <math>\vec{v}=\vec{0}</math> then the equation is always true, and therefore uninteresting.) So now we have | |||
<math>A\vec{v}-\lambda\vec{v}=\vec{0}</math>. It doesn't make sense to subtract a number from a matrix, but we can factor out the vector if we first multiply the right-hand term by the identity, giving us | |||
:<math>(A-\lambda I)\vec{v}=\vec{0}\ .</math> | |||
Now we have to remember the fact that <math>A-\lambda I</math> is a square matrix, and so it might be [[matrix inverse|invertible]]. | |||
If it was invertible then we could simply multiply on the left by its inverse to get | |||
:<math>\vec{v}=(A-\lambda I)^{-1}\vec{0}=\vec{0}</math> | |||
but we have already said that <math>\vec{v}</math> can't be the zero vector! The only way around this is if <math>A-\lambda I</math> is in fact non-invertible. It can be shown that a square matrix is non-invertible if and only if its [[determinant]] is zero. That is, we require | |||
:<math>\text{det}(A-\lambda I)=0\ ,</math> | |||
which is the eigenvalue equation stated above. | |||
Revision as of 18:02, 3 October 2007
In linear algebra an eigenvalue of a (square) 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} is a number 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} that satisfies the eigenvalue 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 \text{det}(A-\lambda I)=0\ ,}
where 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 the identity matrix of the same dimension as 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} and in general 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} can be complex. The origin of this equation is the eigenvalue problem, which is to find the eigenvalues and associated eigenvectors 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} . That is, to find a number 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} and a 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 \vec{v}} that together satisfy
- 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\vec{v}=\lambda\vec{v}\ .}
What this equation says is that even though 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} is a matrix its action on 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 \vec{v}} is the same as multiplying it by the number 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} . This means that the 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 \vec{v}} and the 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 A\vec{v}} are parallel (or anti-parallel if 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} is negative). Note that generally this will not be true. This is most easily seen with a quick example. Suppose
- 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{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}} 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 \vec{v}=\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}\ .}
Then their matrix product 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 A\vec{v}=\begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}\begin{pmatrix} v_1 \\ v_2 \end{pmatrix} =\begin{pmatrix}a_{11}v_1+a_{12}v_2 \\ a_{21}v_1+a_{22}v_2 \end{pmatrix}}
whereas the scalar product 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 \lambda\vec{v}=\begin{pmatrix} \lambda v_1 \\ \lambda v_2 \end{pmatrix}\ .}
Obviously then 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\vec{v}\neq \lambda\vec{v}} unless 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 v_1 = a_{11}v_1+a_{12}v_2} and simultaneously 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 v_2 = a_{21}v_1+a_{22}v_2} , and it is easy to pick numbers for the entries 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} 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 \vec{v}} such that this cannot happen for any value 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 \lambda} .
The eigenvalue equation
So where did the eigenvalue 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 \text{det}(A-\lambda I)=0} come from? Well, we assume that we know the 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} and want to find a number 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} and a non-zero 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 \vec{v}} so that 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\vec{v}=\lambda\vec{v}} . (Note that if 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 \vec{v}=\vec{0}} then the equation is always true, and therefore uninteresting.) So now we have 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\vec{v}-\lambda\vec{v}=\vec{0}} . It doesn't make sense to subtract a number from a matrix, but we can factor out the vector if we first multiply the right-hand term by the identity, giving 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 (A-\lambda I)\vec{v}=\vec{0}\ .}
Now we have to remember the fact that 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} is a square matrix, and so it might be invertible. If it was invertible then we could simply multiply on the left by its inverse to get
- 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 \vec{v}=(A-\lambda I)^{-1}\vec{0}=\vec{0}}
but we have already said that 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 \vec{v}} can't be the zero vector! The only way around this is if 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} is in fact non-invertible. It can be shown that a square matrix is non-invertible if and only if its determinant is zero. That is, we require
- 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 \text{det}(A-\lambda I)=0\ ,}
which is the eigenvalue equation stated above.