VII Hermitian and quadratic forms

31. Transformation of Hermitian forms

A Hermitian form* in the variables x₁, ··· x_n has the form

where the coefficients h_ik are arbitrary, real or complex numbers, which are only subject to the condition . The matrix H = (h_ik) has then the property

and is called a Hermitian matrix. A real Hermitian matrix satisfies the equation H = H', whence it is said to be symmetric. A Hermitian form with real coefficients is a quadratic form; in most cases, you consider quadratic forms only for real values of the variables. **

* Charles Hermite 1822 - 1901
** The following results also apply to quadratic forms in real variables after you omit the conjugate bars on top.

For arbitrary real or complex values of the variables, the value of a Hermitian form is always real. In fact, the value of a Hermitian form does not change by transition to conjugate complex values, for, if you take into consideration that , you find

One of the main tasks of this part is the transformation of Hermitian forms to a much simpler form by introduction of new variables. First of all, we will see how a Hermitian form behaves when new variables are introduced by linear substitution. It is readily confirmed that introduction of the column x with the variables x₁, ··· , x_n yields

Let the new variables y₁, ··· , y_n, denoted as a column by y, be related to x by the regular linear substitution

x = Py. (31.2)

Then and

where the coefficient matrix B is again a Hermitian matrix, because, by (31.1),

As a consequence, you obtain again in the new variables y₁, ··· , y_n a Hermitian form.

In most cases, you do not apply arbitrary, regular variable transformations (31.2), but transformations with a unitary or, in the case of a real quadratic form, a real orthogonal matrix (cf. Sections 18 and 25).

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