Eigenvalue problems
Main idea and formulation in the linear algebra
The word "eigenvalue" stems from the German word "Eigenwert" that can be translated into English as "Its own value" or "Inherent value". This is a value of a parameter in the equation or system of equations for which this equation has a nontriv- ial (nonzero) solution. Mathematically, the simplest formulation of the eigenvalue problem is in the linear algebra. For a given square matrix A one has to find such values of l, for which the equation (actually the system of linear equations) A.X λX (1) has a nontrivial solution for a vector (column) X. Moving the right part to the left, one obtains the equation A − λI .X 0, H L where I is the identity matrix having all diagonal elements one and nondiagonal elements zero. This matrix equation has nontrivial solutions only if its determinant is zero, Det A − λI 0. @ D This is equivalent to a Nth order algebraic equation for l, where N is the rank of the mathrix A. Thus there are N different eigenvalues ln (that can be complex), for which one can find the corresponding eigenvectors Xn. Eigenvectors are defined up to an arbitrary numerical factor, so that usually they are normalized by requiring
T∗ Xn .Xn 1, where X T* is the row transposed and complex conjugate to the column X.
It can be proven that eigenvectors that belong to different eigenvalues are orthogonal, so that, more generally than above, one has
T∗ Xm .Xn δmn .
Here dmn is the Kronecker symbol, 1, m n δ = mn µ 0, m ≠ n.
An important class of square matrices are Hermitean matrices that satisfy
AT∗ A.
Eigenvalues of Hermitean matrices are real. A real Hermitean matrix is just a symmetric matrix, AT ã A. For such matrices eigenvectors can be chosen real.
ü Matrix eigenvalue problem in Mathematica
Mathematica offers a solver for the matrix eivenvalue problem. If one is interested in eigenvalues only, one can use the command Eigenvalues[...]. Eigenvectors are computed by Eigenvectors[...], while both eigenvalues and eigenvectors are computed by the command Eigensystem[...]. Let us illustrate how it works for a real symmetric matrix
a b A = ; K b − a O
Its eigenvalues are given by 2 Mathematical_physics-14-Eigenvalue problems.nb
Eigenvalues A @ D
− a2 + b2 , a2 + b2 : >
Its eigenvectors are given by
Eigenvectors A @ D
−a + a2 + b2 −a − a2 + b2 − , 1 , − , 1 :: b > : b >> that is,
X := Eigenvectors A i i_ @ D@@ DD These two eigenvectors are orthogonal to each other
X .X Simplify 1 2 êê 0
However, they are not normalized
X .X Expand Factor 1 1 êê êê
2 −a2 − b2 + a a2 + b2 − b2
To see which eigenvector corresponds to each eigenvalue, one has to use the command Eigensystem
ESys = Eigensystem A @ D
− a + a2 + b2 −a − a2 + b2 − a2 + b2 , a2 + b2 , − , 1 , − , 1 :: > :: b > : b >>>
The first part of this List are eigenvalues and the second part are eigenvectors. One can better see the correspondence in the form
TableForm Transpose ESys @ @ DD
−a+ a2+b2 2 2 − − a + b b 1
−a− a2+b2 2 2 − a + b b 1
Mathematica also solves matrix eigenvalue problems numerically, that is the only way to go for big matrices. For instance,
ESys = Eigensystem A . a → 1., b → 2. @ ê 8 The numerical eigenvectors X := ESys 2 i i_ @@ DD@@ DD are orthonormal Mathematical_physics-14-Eigenvalue problems.nb 3 X1.X1 X1.X2 1. 0. For a complex Hermitean matrix eigenvalues are indeed real, although eigenvectors are complex 2 TableForm Transpose Eigensystem B B BK − 2 OFFF