Generalized eigenvalue problem

The generalized eigenvalue problem is a problem of linear algebra .

definition

The problem at predetermined matrices certain numbers and vectors to determine to so ${\displaystyle A,B\in \mathbb {C} ^{n\times n}}$ ${\displaystyle \lambda \in \mathbb {C} }$ ${\displaystyle x\in \mathbb {C} ^{n}}$ ${\displaystyle x\neq 0}$

${\displaystyle Ax=\lambda Bx}$

applies, is referred to as a generalized eigenvalue problem as a distinction to the eigenvalue problem .

Solution method

If regular , then the generalized eigenvalue problem can be applied to the ordinary eigenvalue problem ${\displaystyle B}$

${\displaystyle B^{-1}Ax=\lambda x}$

lead back. But this approach is i. A. only of theoretical importance, since the calculation of an inverse matrix is often not numerically possible or very impractical. Often certain information about the observed matrices can already be collected from the task, which can then simplify the calculation. Are z. B. symmetrically and also positive definite , so the calculation can be simplified significantly: The matrix can be by means of Cholesky decomposition in decompose. Then is similar to a matrix . The inverse of can be calculated very efficiently because is a triangular matrix. If one now determines the eigenvalues ​​of , then these are also the eigenvalues ​​of . ${\displaystyle A,B}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle B=LL^{T}}$ ${\displaystyle B^{-1}A}$ ${\displaystyle H:=L^{-1}A(L^{-1})^{T}}$ ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle H}$ ${\displaystyle B^{-1}A}$

The QZ algorithm can also be used for any matrices . ${\displaystyle A,B}$

example

Consider the generalized eigenvalue problem

${\displaystyle {\begin{bmatrix}4&4&8\\4&0&4\\8&4&4\end{bmatrix}}x=\lambda {\begin{bmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{bmatrix}}x}$.

Naive approach

Calculating the inverse of gives ${\displaystyle B}$

${\displaystyle B^{-1}={\frac {1}{4}}{\begin{bmatrix}3&2&1\\2&4&2\\1&2&3\end{bmatrix}}}$

and thus

${\displaystyle B^{-1}A={\begin{bmatrix}7&4&9\\10&4&10\\9&4&7\end{bmatrix}}}$.

The eigenvalues ​​of this matrix are 20.7703 as well as -2 and - 0.7703.

Using the Cholesky decomposition

${\displaystyle A,B}$are symmetrical and also positive definite. The Cholesky decomposition provides the matrix ${\displaystyle B}$

${\displaystyle L^{T}={\begin{bmatrix}1{,}4142&-0{,}7071&0\\0&1{,}2247&-0{,}8165\\0&0&1{,}1547\end{bmatrix}}}$.

Then is . ${\displaystyle H:=L^{-1}A(L^{-1})^{T}={\begin{bmatrix}2{,}0000&3{,}4641&7{,}3485\\3{,}4641&3{,}3333&8{,}0139\\7{,}3485&8{,}0139&12{,}6667\end{bmatrix}}}$

As expected, the eigenvalues ​​of this matrix are identical to the eigenvalues ​​calculated above.

literature

• Peter Knabner , Wolf Barth : Lineare Algebra . Basics and applications (=  Springer textbook ). Springer, Berlin 2012, ISBN 978-3-642-32185-6 .
• Josef Stoer, Roland Bulirsch : Numerical Mathematics 2nd 5th edition, Springer, Berlin / Heidelberg / New York 2005, ISBN 978-3-540-23777-8 .