33. Principal axes transformation

The statement that a Hermitian form can always be given the form (32.1) by introduction of suitable variables becomes in terms of matrices

Theorem 19

There corresponds to every Hermitian matrix H a unitary matrix U with the property that is a diagonal matrix. Along the principal diagonal of C occur the characteristic roots of H.

Proof: Employ induction with the number of rows n of H. No proof is required for n = 1. Assume that the rule has been proved for Hermitian matrices with less than n rows.

Now consider a Hermitian (n,n )-matrix. Denote a characteristic root by g₁ and its eigenvector by x₁. Since x₁ is obtained as a solution of a homogeneous system of equations, you can assume that it is normalized, that is, = 1.

Construct a unitry matrix U₁ the first column of which is . You can do this because, by Section 27, the vector space of all vectors orthogonal to has the dimension n - 1, whence there exists for this vector space a base of n - 1 pairs of normalized vectors ₂, ··· , _n for which

Together with = 1, these equations describe the matrix U₁ with the columns , ₂ , ··· , _n as a unitary matrix. In order to indicate that U₁ consists of these columns, write

U₁ = (, ₂ , ··· , _n).

Then has the rows so that

In order to form the product compute first of all HU₁. Since as eigenvector of H of the equation satisfies

Moreover

where on the right hand side only the elements of the first row have been written down. Since = 1, the top element of the first column equals g₁ and due to the first equation (33.1) all other elements of the first column vanish. Since now like H is Hermitian, the elements, at positions reflected in the main diagonal, must be conjugate complex, whence in also the second to the n-th elements of the first row vanish and

with a certain (n - 1,n - 1)-matrix H₁. Since is Hermitian, obviously also H₁ is Hermitian. According to the assumption of the induction, there exists a unitary (n - 1,n - 1)-matrix U₂. with the property that becomes a diagonal matrix:

The (n,n)--matrix

,

as is readily seen, is also unitary. Moreover, as a product of two such matrices, U = U₁U₂ is again a unitary matrix . This matrix employs now the property, stated in Theorem 19, because

whence the rule has been proved.

We yet make an observation regarding the principal axes transformation of real symmetric matrices. In that case, everything happens in the same manner except that all the matrices ar real and therefore the bars, indicating transition to conjugte complex numbers, can be omitted and the unitary matrices are replaced by real, orthogonal matrices. Everything takes place in the real domain, because all the characteristic roots of a real, symmetric matrix as well as those of a Hermitian matrix are real (Theorem 17). As a consequence, the eigenvector , employed above, can be real and you can construct a real, orthogonal matrix U with as first column.

We can still say more regarding the matrices U, which yield the principal axes transformation then becomes clear during the induction proof above. First of all, let all the characteristic roots g₁, ··· , g_n of H be different. Then there exist n different eigenvectors , ··· , , which by Theorem 18 are pairwise orthogonal and can be assumed to have been normalized. The matrix

with the columns , ··· , is then unitary and has the properties, stated in Theorem 19, because

A Matrix U, which performs the principal axes transformation for H, can therefore in this case be constructed entirely from the eigenvectors of H. It is not straight forward to see that such a construction is also possible in the general case when not all of the characteristic roots of H differ, because it is not known, whether there are as before n linearly independent eigenvectors of H available. In fact, this is due to the property of Hermitian matrices:

If g is an s-fold characteristic root of the Hermitian matrix H, then H has exactly s linearly independent eigenvectors which belong to g .

All the eigenvectors belonging to g are solutions of the homogeneous system of equations

By Theorem 14, this has exactly n - r linearly independent solutions, where r is the Rank of gE - H. If you employ the principal axes transformation, this Rank can be computed: Find a unitary matrix U, for which becomes a diagonal matrix. Then

Now, on the one hand, according to the Rank equation in Section 24,

on the other hand, the Rank of the diagonal matrix is n - s, where s is the number of zeroes in the principal diagonal or, what is the same, is the multiplicity of the charcteristic roots g. Hence

r = Rank(gE - H) = n - s

and the system of equations (33.2) has indeed

n - (n - s) = s

linearly independent solutions.

Since there exist for every characteristic root as many linearly independent eigenvectors as is determined by their multiplicity, the Hermitian matrix H has in each case n linearly independent eigenvectors. According to Theorem 18, the eigenvectors, belonging to different characteristic roots, are orthogonal; those belonging to one and the same characteristic root are to be orthogonalized by the method of Section 25. In this manner, you obtain a normalized, orthogonal system of n eigenvectors of H. A unitary matrix U with these eigenvectors as columns yields the principal axes transformation for H, as can be confirmed as above in the case of all different characteristic roots. *

* Beware of a repetitive conclusion! The statement regarding the number of linearly independent eigenvectors in the case of a multiple characteristic root has been proved by means of the principal axes transformation. If you want to employ this statement conversely, you must perform a different proof which does not employ the principal axes of transformation.

It may happen that the determinant |H| vanishes for a Hermitian matrix H. If the Rank of H equals r, the Hermitian form can be rewriten in terms of variables, which are related to x by a unitary transformation, as a form in r variables. Since

Rank H = Rank 18 = r, (33.3)

exactly r elements among the diagonal elements of the matrix C = are non-zero, so that indeed there occur in (32.1) only r of the variables y₁, ··· , y_n. On the other hand, by (33.3), the form can also not be written as a form of less than r variables.

If you do not restrict consideration to unitary transformations of the variables, but admit arbitrary, non-singular substitutions, you can by introduction of suitable variables achieve a yet simpler version of Hermitian form. In every case, you can reach by a unitary transformation the form (32.1). Let the rank of H be r and renumber the variables so that g_r+1= ··· = g_n = 0. In this form

g₁|y₁|² + ··· + g_r|y_r|², (33.4)

let

Complete this new (in general, non-unitary) transformation by

z_k = y_k for k = r +1, ··· , n

into a non-singular substitution of all n variables when (33.4) becomes

e₁|z₁|² + ··· + e_r|z_r|² with e_i = ± 1.

The transformation of variables, yielding this form, is by no means uniquely determined. For example, let

be achieved by two different transformations and the transition substitution be

Then there applies the so called Inertia Rule of Hermitian forms: The number of positive e_i equals the number of positive h_i. We present an indirect proof by assuming that

e₁ = ··· = e_p = + 1, e_p+1 = ··· = e_r = -1,

h₁ = ··· = h_q = + 1, e_q+1 = ··· = h_r = + 1

and p < q. Now try to find the variables x so that

z₁ = ··· = z_p = 0, (33.7)
v_q+1 = ··· = v_r = 0. (33.8)

These are p + r - q + n - r n equations. Then

and due to (33.8)

which is obviously impossible.

last next