One of the most useful concepts in the theory of normal transformations on unitary spaces is that of a function of a transformation. If \(A\) is a normal transformation with spectral form \(\sum_{j} \alpha_{j} E_{j}\) (for this discussion we temporarily assume that the underlying vector space is a unitary space), and if \(f\) is an arbitrary complex-valued function defined at least at the points \(\alpha_{j}\) , then we define a linear transformation \(f(A)\) by \[f(A)=\sum_{j} f(\alpha_{j}) E_{j}.\] Since for polynomials \(p\) (and even for rational functions) we have already seen that our earlier definition of \(p(A)\) yields, if \(A\) is normal, \(p(A)=\sum_{j} p(\alpha_{j}) E_{j}\) , we see that the new notion is a generalization of the old one. The advantage of considering \(f(A)\) for arbitrary functions \(f\) is for us largely notational; it introduces nothing conceptually new. Indeed, for an arbitrary \(f\) , we may write \(f(\alpha_{j})=\beta_{j}\) , and then we may find a polynomial \(p\) that at the finite set of distinct complex numbers \(\alpha_{j}\) takes, respectively, the values \(\beta_{j}\) . With this polynomial \(p\) we have \(f(A)=p(A)\) , so that the class of transformations defined by the formation of arbitrary functions is nothing essentially new; it only saves the trouble of constructing a polynomial to fit each special case. Thus for example, if, for each complex number \(\lambda\) , we write \[f_{\lambda}(\zeta)=0 \quad \text{ whenever } \quad \zeta \neq \lambda\] and \[f_{\lambda}(\lambda)=1,\] then \(f_{\lambda}(A)\) is the perpendicular projection on the subspace of solutions of \(A x=\lambda x\) .

We observe that if \(f(\zeta)=\frac{1}{\zeta}\) , then (assuming of course that \(f\) is defined for all \(\alpha_{j}\) , that is, that \(\alpha_{j} \neq 0\) ) \(f(A)=A^{-1}\) , and if \(f(\zeta)=\xi\) , then \(f(A)=A^{*}\) . These statements imply that if \(f\) is an arbitrary rational function of \(\zeta\) and \(\xi\) , we obtain \(f(A)\) by the replacements \(\zeta \to A\) , \(\xi \to A^{*}\) , and \(\frac{1}{\xi}=A^{-1}\) . The symbol \(f(A)\) is, however, defined for much more general functions, and in the sequel we shall feel free to make use of expressions such as \(e^{A}\) and \(\sqrt{A}\) .

A particularly important function is the square root of positive transformations. We consider \(f(\zeta)=\sqrt{\zeta}\) , defined for all real \(\zeta \geq 0\) , as the positive square root of \(\zeta\) , and for every positive \(A=\sum_{j} \alpha_{j} E_{j}\) we write \[\sqrt{A}=\sum_{j} \sqrt{\alpha_{j}} E_{j}.\] (Recall that \(\alpha_{j} \geq 0\) for all \(j\) . The discussion that follows applies to both real and complex inner product spaces.) It is clear that \(\sqrt{A} \geq 0\) and that \((\sqrt{A})^{2}=A\) ; we should like to investigate the extent to which these properties characterize \(\sqrt{A}\) . At first glance it may seem hopeless to look for any uniqueness, since if we consider \(B=\sum_{j} \pm \sqrt{\alpha_{j}} E_{j}\) , with an arbitrary choice of sign in each place, we still have \(A=B^{2}\) . The transformation \(\sqrt{A}\) that we constructed, however, was positive, and we can show that this additional property guarantees uniqueness. In other words: if \(A=B^{2}\) and \(B \geq 0\) , then \(B=\sqrt{A}\) . To prove this, let \(B=\sum_{k} \beta_{k} F_{k}\) be the spectral form of \(B\) ; then \[\sum_{k} \beta_{k}^{2} F_{k}=B^{2}=A=\sum_{j} \alpha_{j} E_{j}.\] Since the \(\beta_{k}\) are distinct and positive, so also are the \(\beta_{k}^{2}\) ; the uniqueness of the spectral form of \(A\) implies that each \(\beta_{k}^{2}\) is equal to some \(\alpha_{j}\) (and vice versa), and that the corresponding \(E\) ’s and \(F\) ’s are equal. By a permutation of the indices we may therefore achieve \(\beta_{j}^{2}=\alpha_{j}\) for all \(j\) , so that \(\beta_{j}=\sqrt{\alpha_{j}}\) , as was to be shown.

There are several important applications of the existence of square roots for positive operators; we shall now give two of them.

First: we recall that in Section: Positive transformations we mentioned three possible definitions of a positive transformation \(A\) , and adopted the weakest one, namely, that \(A\) is self-adjoint and \((A x, x) \geq 0\) for all \(x\) . The strongest of the three possible definitions was that we could write \(A\) in the form \(A=B^{2}\) for some self-adjoint \(B\) . We point out that the result of this section concerning square roots implies that the (seemingly) weakest one of our conditions implies and is therefore equivalent to the strongest. (In fact, we can even obtain a unique positive square root.)

Second: in Section: Positive transformations we stated also that if \(A\) and \(B\) are positive and commutative, then \(A B\) is also positive; we can now give an easy proof of this assertion. Since \(\sqrt{A}\) and \(\sqrt{B}\) are functions of (polynomials in) \(A\) and \(B\) respectively, the commutativity of \(A\) and \(B\) implies that \(\sqrt{A}\) and \(\sqrt{B}\) commute with each other; consequently \begin{align} A B &= \sqrt{A} \sqrt{A} \sqrt{B} \sqrt{B}\\ &= \sqrt{A} \sqrt{B} \sqrt{A} \sqrt{B}\\ &= (\sqrt{A} \sqrt{B})^{2}. \end{align} Since \(\sqrt{A}\) and \(\sqrt{B}\) are self-adjoint and commutative, their product is self-adjoint and therefore its square is positive.

Spectral theory also makes it quite easy to characterize the matrix (with respect to an arbitrary orthonormal coordinate system) of a positive transformation \(A\) . Since \(\operatorname{det} A\) is the product of the proper values of \(A\) , it is clear that \(A \geq 0\) implies \(\det A \geq 0\) . (The discussion in Section: Multiplicity applies directly to complex inner product spaces only; the appropriate modification needed for the discussion of self-adjoint transformations on possibly real spaces is, however, quite easy to supply.) If we consider the defining property of positiveness expressed in terms of the matrix \((\alpha_{i j})\) of \(A\) , that is, \(\sum_{i} \sum_{j} \alpha_{i} \xi_{i} \bar{\xi}_{j} \geq 0\) , we observe that the last expression remains positive if we restrict the coordinates \((\xi_{1}, \ldots, \xi_{n})\) by requiring that certain ones of them vanish. In terms of the matrix this means that if we cross out the columns numbered \(j_{1}, \ldots, j_{k}\) , say, and cross out also the rows bearing the same numbers, the remaining small matrix is still positive, and consequently so is its determinant. This fact is usually expressed by saying that the principal minors of the determinant of a positive matrix are positive. The converse is true. The coefficient of the \(j\) -th power of \(\lambda\) in the characteristic polynomial \(\operatorname{det}(A-\lambda)\) of \(A\) is (except for sign) the sum of all principal minors of \(n-j\) rows and columns. The sign is alternately plus and minus; this implies that if \(A\) has positive principal minors and is self-adjoint (so that the zeros of \(\det(A-\lambda)\) are known to be real), then the proper values of \(A\) are positive. Since the self-adjoint character of a matrix is ascertainable by observing whether or not it is (Hermitian) symmetric ( \(\alpha_{i j}=\bar{\alpha}_{j i}\) ), our comments reduce the problem of finding out whether or not a matrix is positive to a finite number of elementary computations.

EXERCISES