Mathematical Methods for Physics and Engineering - Matematica.NET

8.13 EIGENVECTORS AND EIGENVALUESrepresent the eigenvectors x of A in our chosen coordinate system. Conventionally,these column matrices are also referred to as the eigenvectors of the matrixA. § Clearly, if x is an eigenvector of A (with some eigenvalue λ) then any scalarmultiple µx is also an eigenvector with the same eigenvalue. We therefore oftenuse normalised eigenvectors, for whichx † x =1(note that x † x corresponds to the inner product 〈x|x〉 in our basis). Any eigenvectorx can be normalised by dividing all its components by the scalar (x † x) 1/2 .As will be seen, the problem of finding the eigenvalues and correspondingeigenvectors of a square matrix A plays an important role in many physicalinvestigations. Throughout this chapter we denote the ith eigenvector of a squarematrix A by x i and the corresponding eigenvalue by λ i . This superscript notationfor eigenvectors is used to avoid any confusion with components.◮A non-singular matrix A has eigenvalues λ i and eigenvectors x i . Find the eigenvalues andeigenvectors of the inverse matrix A −1 .The eigenvalues and eigenvectors of A satisfyAx i = λ i x i .Left-multiplying both sides of this equation by A −1 , we findA −1 Ax i = λ i A −1 x i .Since A −1 A = I, on rearranging we obtainA −1 x i = 1 λ ix i .Thus, we see that A −1 has the same eigenvectors x i as does A, but the correspondingeigenvalues are 1/λ i . ◭In the remainder of this section we will discuss some useful results concerningthe eigenvectors and eigenvalues of certain special (though commonly occurring)square matrices. The results will be established for matrices whose elements maybe complex; the corresponding properties for real matrices may be obtained asspecial cases.8.13.1 Eigenvectors and eigenvalues of a normal matrixIn subsection 8.12.7 we defined a normal matrix A as one that commutes with itsHermitian conjugate, so thatA † A = AA † .§ In this context, when referring to linear combinations of eigenvectors x we will normally use theterm ‘vector’.273