By Hans F. Weinberger
Offers a typical atmosphere for varied equipment of bounding the eigenvalues of a self-adjoint linear operator and emphasizes their relationships. A mapping precept is gifted to attach a few of the equipment. The eigenvalue difficulties studied are linear, and linearization is proven to provide very important information regarding nonlinear difficulties. Linear vector areas and their houses are used to uniformly describe the eigenvalue difficulties awarded that contain matrices, traditional or partial differential operators, and integro-differential operators.
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5 as a criterion to establish the existence of eigenvalues, we need a way of generating linear functionals with the needed properties. The most important result along these lines is the following. 7 (Poincare's inequality). Let iKx^ • • • , XN) be a square integrable function with square integrable first partial derivatives on the N -dimensional cube Ka of side a(ve Hl(Ka)). Then a Proof. For the sake of simplicity, we shall prove the theorem in two dimensions. The ideas are the same in one dimension or in more dimensions.
DEFINITION. A mapping from a linear vector space Fto the real numbers is said to be a quadratic functional (or form) if there exists a Hermitian (or, if F is over the field of real numbers, a symmetric) functional B(u, v) on F x F in terms of which the mapping has the form u -> B(u, u). Note that if B is Hermitian, then B(u, u) = B(u, u) and hence B(u, u) is real. 1) implies that so that the Hermitian form B(u, v) can be recovered from a knowledge of the quadratic form B(u, u) for all u in F. Thus, there is a one-to-one correspondence between Hermitian sesquilinear functionals and quadratic functionals.
On the other hand, since the conditions &f(v, u , ) = • • • = ,c/(r, w v ) = 0 imply v = 0, ^n is not defined for n > N. 1 require t h a t A and B be represented by Hermitian matrices and that A be positive definite. 4) that the unit matrix, and hence that a diagonal matrix. 3. COROLLARY. // A and B are two Hermitian N x N matrices and A is positive definite, there is a nonsingular matrix Q with the properties that Q*AQ is the unit matrix and Q*BQ is a real diagonal matrix. The diagonal entries of Q*BQ are the eigenvalues of the problem Bu = iiAu.