Notes on Eigenvalues, Singular Values and QR

for some nonzero vector x called an eigenvector. This is equivalent to writing (λI − A)x = 0 so, since x 6= 0, A− λI must be singular, and hence det(λI − A) = 0. From the (complicated!) definition of determinant, it follows that det(λI−A) is a polynomial in the variable λ with degree n, and this is called the characteristic polynomial. By the fundamental theorem of algebra (a nontrivial result), it follows that the characteristic polynomial has n roots which we denote λ1, . . . , λn, but these may not be distinct (different from each other). For example, the identity matrix I has characteristic polynomial (λ−1) and so all its eigenvalues are equal to one. Note that if A is real, the eigenvalues may not all be real, but those that are not real must occur in complex conjugate pairs λ = α±βi. It does not matter what order we use for numbering the λj. Although in principle we could compute eigenvalues by finding the roots of the characteristic polynomial, in practice there are much better algorithms, and in any case there is no general formula for finding the roots If x is an eigenvector, so is αx for any nonzero scalar α.