8.13 EIGENVECTORS AND EIGENVALUES
represent the eigenvectorsxofAin our chosen coordinate system. Convention-
ally, these column matrices are also referred to as theeigenvectors of the matrix
A.§Clearly, ifxis an eigenvector ofA(with some eigenvalueλ) then any scalar
multipleμxis also an eigenvector with the same eigenvalue. We therefore often
usenormalisedeigenvectors, for which
x†x=1(note thatx†xcorresponds to the inner product〈x|x〉in our basis). Any eigen-
vectorxcan 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 matrixAplays an important role in many physical
investigations. Throughout this chapter we denote theith eigenvector of a square
matrixAbyxiand the corresponding eigenvalue byλi. This superscript notation
for eigenvectors is used to avoid any confusion with components.
A non-singular matrixAhas eigenvaluesλiand eigenvectorsxi. Find the eigenvalues and
eigenvectors of the inverse matrixA−^1.The eigenvalues and eigenvectors ofAsatisfy
Axi=λixi.Left-multiplying both sides of this equation byA−^1 , we find
A−^1 Axi=λiA−^1 xi.SinceA−^1 A=I, on rearranging we obtain
A−^1 xi=1
λixi.Thus, we see thatA−^1 has thesameeigenvectorsxias doesA, but the corresponding
eigenvalues are 1/λi.
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 may
be complex; the corresponding properties for real matrices may be obtained as
special cases.
8.13.1 Eigenvectors and eigenvalues of a normal matrixIn subsection 8.12.7 we defined a normal matrixAas one that commutes with its
Hermitian conjugate, so that
A†A=AA†.§In this context, when referring to linear combinations of eigenvectorsxwe will normally use the
term ‘vector’.