Matrix eigenvalue theory

Any column vector which satisfies this equation is called an

Equation (A.144) can be rearranged to give

(A.145) |

where is the unit matrix. The preceding matrix equation is essentially a set of homogeneous simultaneous algebraic equations for the components of . A well-known property of such a set of equations is that it only has a nontrivial solution when the determinant of the associated matrix is set to zero (Gradshteyn and Ryzhik 1980c). Hence, a necessary condition for the preceding set of equations to have a nontrivial solution is that

(A.146) |

where denotes a determinant. This formula is essentially an th-order polynomial equation for . We know that such an equation has (possibly complex) roots. Hence, we conclude that there are eigenvalues, and associated eigenvectors, of the -dimensional matrix .

Let us now demonstrate that the eigenvalues and eigenvectors of the real symmetric matrix are all real. We have

and, taking the transpose and complex conjugate,

where and are the th eigenvector and eigenvalue of , respectively. Left multiplying Equation (A.147) by , we obtain

(A.149) |

Likewise, right multiplying Equation (A.148) by , we get

(A.150) |

The difference of the previous two equations yields

(A.151) |

It follows that , because (which is in vector notation) is real and positive definite. Hence, is real. It immediately follows that is real.

Next, let us show that two eigenvectors corresponding to two different eigenvalues are mutually orthogonal. Let

(A.152) | ||||||

and | (A.153) |

where . Taking the transpose of the first equation and right multiplying by , and left multiplying the second equation by , we obtain

(A.154) | ||||||

and | (A.155) |

Taking the difference of the preceding two equations, we get

(A.156) |

Because, by hypothesis, , it follows that . In vector notation, this is the same as . Hence, the eigenvectors and are mutually orthogonal.

Suppose that . In this case, we cannot conclude that by the preceding argument. However, it is easily seen that any linear combination of and is an eigenvector of with eigenvalue . Hence, it is possible to define two new eigenvectors of , with the eigenvalue , which are mutually orthogonal. For instance,

(A.157) | ||||||

and | (A.158) |

It should be clear that this argument can be generalized to deal with any number of eigenvalues that take the same value.

In conclusion, a real symmetric -dimensional matrix possesses real eigenvalues, with associated real eigenvectors, which are, or can be chosen to be, mutually orthogonal.