Min-max theorem

rdf:langString Min-max theorem

rdf:langString Since the matrix is Hermitian it is diagonalizable and we can choose an orthonormal basis of eigenvectors {u1, ..., un} that is, ui is an eigenvector for the eigenvalue λi and such that = 1 and = 0 for all i ≠ j. If U is a subspace of dimension k then its intersection with the subspace isn't zero, for if it were, then the dimension of the span of the two subspaces would be , which is impossible. Hence there exists a vector in this intersection that we can write as : and whose Rayleigh quotient is : and hence : Since this is true for all U, we can conclude that : This is one inequality. To establish the other inequality, chose the specific k-dimensional space , for which : because is the largest eigenvalue in V. Therefore, also : To get the other formula, consider the Hermitian matrix , whose eigenvalues in increasing order are . Applying the result just proved, : The result follows on replacing with .

rdf:langString Let S' be the closure of the linear span . The subspace S' has codimension k − 1. By the same dimension count argument as in the matrix case, S' ∩ Sk is non empty. So there exists x ∈ S' ∩ Sk with . Since it is an element of S' , such an x necessarily satisfy : Therefore, for all Sk : But is compact, therefore the function f = is weakly continuous. Furthermore, any bounded set in H is weakly compact. This lets us replace the infimum by minimum: : So : Because equality is achieved when , : This is the first part of min-max theorem for compact self-adjoint operators. Analogously, consider now a -dimensional subspace S'k−1, whose the orthogonal complement is denoted by S'k−1⊥. If S' = span{u1...uk}, : So : This implies : where the compactness of A was applied. Index the above by the collection of k-1-dimensional subspaces gives : Pick S'k−1 = span{u1, ..., u'k−1} and we deduce :

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rdf:langString In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature. This article first discusses the finite-dimensional case and its applications before considering compact operators on infinite-dimensional Hilbert spaces. We will see that for compact operators, the proof of the main theorem uses essentially the same idea from the finite-dimensional argument. In the case that the operator is non-Hermitian, the theorem provides an equivalent characterization of the associated singular values. The min-max theorem can be extended to self-adjoint operators that are bounded below.