Characterization of spectra

The following results support the analogy between numbers and transformations more than anything so far; they assert that the properties that caused us to define the special classes of transformations we have been considering are reflected by their spectra.

Theorem 1. If is a self-adjoint transformation on an inner product space, then every proper value of is real; if is positive, or strictly positive, then every proper value of is positive, or strictly positive, respectively.

Proof. We may ignore the fact that the first assertion is trivial in the real case; the same proof serves to establish both assertions in both the real and the complex case. Indeed, if , with , then, it follows that if is real (see Section: Polarization , Theorem 4), then so is , and if is positive (or strictly positive) then so is . ◻

Theorem 2. Every root of the characteristic equation of a self-adjoint transformation on a finite-dimensional inner product space is real.

Proof. In the complex case roots of the characteristic equation are the same thing as proper values, and the result follows from Theorem 1. If is a symmetric transformation on a Euclidean space, then its complexification is Hermitian, and the result follows from the fact that and have the same characteristic equation. ◻

We observe that it is an immediate consequence of Theorem 2 that a self-adjoint transformation on a finite-dimensional inner product space always has a proper value.

Theorem 3. Every proper value of an isometry has absolute value one.

Proof. If is an isometry, and if , with , then . ◻

Theorem 4. If is either self-adjoint or isometric, then proper vectors of belonging to distinct proper values are orthogonal.

Proof. Suppose , , . If is self-adjoint, then (The middle step makes use of the self-adjoint character of , and the last step of the reality of .) In case is an isometry, (1) is replaced by recall that . In either case would imply that , so that we must have . ◻

Theorem 5. If a subspace is invariant under an isometry on a finite-dimensional inner product space, then so is .

Proof. Considered on the finite-dimensional subspace , the transformation is still an isometry, and, consequently, it is invertible. It follows that every in may be written in the form with in ; in other words, if is in and if , then is in . Hence is invariant under . It follows from Section: Adjoints of projections , Theorem 2, that is invariant under . ◻

We observe that the same result for self-adjoint transformations (even in not necessarily finite-dimensional spaces) is trivial, since if is invariant under , then is invariant under .

Theorem 6. If is a self-adjoint transformation on a finite-dimensional inner product space, then the algebraic multiplicity of each proper value of is equal to its geometric multiplicity, that is, to the dimension of the subspace of all solutions of .

Proof. It is clear that is invariant under , and therefore so is ; let us denote by and the linear transformation considered only on and respectively. We have for all . Since is a self-adjoint transformation on a finite-dimensional space, with only one proper value, namely, , it follows that must occur as a proper value of with algebraic multiplicity equal to the dimension of . If that dimension is , then . Since, on the other hand, is not a proper value of at all, and since, consequently, , we see that contains as a factor exactly times, as was to be proved. ◻

What made this proof work was the invariance of and the fact that every root of the characteristic equation of is a proper value of . The latter assertion is true for every linear transformation on a unitary space; the following result is a consequence of these observations and of Theorem 5.

Theorem 7. If is a unitary transformation on a finite-dimensional unitary space, then the algebraic multiplicity of each proper value of is equal to its geometric multiplicity.

EXERCISES

Exercise 1. Give an example of a linear transformation with two non-orthogonal proper vectors belonging to distinct proper values.

Exercise 2. Give an example of a non-positive linear transformation (on a finite-dimensional unitary space) all of whose proper values are positive.

Exercise 3.

If is self-adjoint, then is real.
If is unitary, then .
What can be said about the determinant of a partial isometry?

EXERCISES

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