32. Characteristic roots. Eigenvectors.

Introducing into the Hermitian form new variables by the linear transformation x = Uy with the unitary matrix U, you obtain

where C is again a Hermitian matrix. By appropriate choice of U, you can give a very simple form. Now select the matrix U so that

that is, it becomes a diagonal matrix, whence you find

The transformation by which a Hermitian form becomes so simple that only quadratic terms appear is called a principal axes transformation *. In the case of a real quadratic form, you have, of course, on transformation

g₁y₁² + ··· + g_ny_n² .

* This term reminds us of the principal axes transformation of curves and surfaces of second order which involves also a quadratic form or an orthogonal transformation (rotation) so that only quadratic terms remain.

Prior to confirming the possibility of a principal axes transformation, we will take a closer look at the diagonal elements of the matrix C and try to establish its link to the matrix H. For this purpose, we form with the unknown x the equation

|xE - H| = 0, (32.2)

the so-called characteristic equation of the matrix H. The detailed form of this equation is

We see that the coefficient p_k is the sum of all so-called principal minor determinants of H with k rows, that is, the sum of all those minor determinants of H of order k, the principal diagonals of which are parts of the principal diagonal of H. It is readily confirmed that especially

p₁ = h₁₁ + h₂₂ + ··· + h_nn,

that it is the sum of the principal diagonal elements of H. You also realize immediately by setting x = 0 that p_n = |H|. The roots of the characteristic equation are called the characteristic roots or eigenvalues of H.

Since U is the unitary matrix of the equation , you find

and by formation of the determinant

You see now that the characteristic equations of H and and hence also the characteristic roots of these matrices agree. These roots of the diagonal matrix C are now simply its diagonal elements, because

Hence the coefficients g₁, ··· ,g_n in (32.1) are the characteristic roots of H and you realize that except for their sequence they are determined uniquely by H.

You substitute often in (32.2) for x a characteristic root g_i of H, then |g_nE - H| = 0, whence the homogeneous system of equations

(g_iE - H)x = 0

has a non-trivial solution x = x_i. You call x_i the eigenvector of H, corresponding to the characteristic root g_i.

We will now prove

Theorem 17

All the characteristic roots of a Hermitian matrix are real.

If you want to use the possibility of the principal axes transformation, Theorem 17 follows simply from the fact that the g₁, ··· ,g_n in (32.1) are the characteristic roots of H and the expression (32.1) must be real for every choice of y₁,···,y_n. Without employment of this transformation, you reason as follows: Find for the characteristic root g_i of H its eigenvector x_i. Then (g_iE - H)x_i = 0 or g_ix_i=Hx_i. Multiplying the last equation from the left by , you obtain . The product is a positive number, in fact, it is the norm of (cf. Section 25). On the right hand side of the last equation is a real number, namely the value of the Hermitian form when you substitute for the variables the elements of the eigenvector , whence also g_i must be real.

As regards the eigenvectors, which belong to different eigenvalues, you have

Theorem 18

Two eigenvectors of a Hermitian matrix, which belong to different eigenvalues, are mutually orthogonal.

Proof: Let g_i and g_j be two different characteristic roots of the hermitian matrix H, x_i and x_j be the associated eigenvectors, whence g_ix_i= Hx_i, g_jx_i= Hx_j. Since g_i is real and , you find

Since g_i - g_j 0, you have = 0, as was to be proved.

last next