Minimax principle

Bounds of a self-adjoint transformationConvergence of linear transformations

A very elegant and useful fact concerning self-adjoint transformations is the following minimax principle .

Theorem 1. Let be a self-adjoint transformation on an -dimensional inner product space , and let be the (not necessarily distinct) proper values of , with the notation so chosen that . If, for each subspace of , and if, for , then for .

Proof. Let be an orthonormal basis in for which , ( Section: Spectral theorem ); let be the subspace spanned by , for . Since the dimension of is , the subspace cannot be disjoint from any -dimensional subspace in ; if is any such subspace, we may find a vector belonging to both and and such that . For this we have so that .

If, on the other hand, we consider the particular -dimensional subspace spanned by , then, for each in this subspace, we have (assuming ) so that .

In other words, as runs over all -dimensional subspaces, is always , and is at least once ; this shows that , as was to be proved. ◻

In particular for we see (using Section: Bounds of a self-adjoint transformation ) that if is self-adjoint, then is equal to the maximum of the absolute values of the proper values of .

EXERCISES

Exercise 1. If is a proper value of a linear transformation on a finite-dimensional inner product space, then .

Exercise 2. If and are linear transformations on a finite-dimensional unitary space, and if , then . (Hint: consider the proper values of .)

Exercise 3. If and are linear transformations on a finite-dimensional unitary space, if , and if commutes with , then is not invertible. (Hint: if is invertible, then .)

Exercise 4. 

  1. If is a normal linear transformation on a finite-dimensional unitary space, then is equal to the maximum of the absolute values of the proper values of .
  2. Does the conclusion of (a) remain true if the hypothesis of normality is omitted?

Exercise 5. The spectral radius of a linear transformation on a finite-dimensional unitary space, denoted by , is the maximum of the absolute values of the proper values of .

  1. If , then is an analytic function of in the region determined by (for each fixed and ).
  2. There exists a constant such that whenever and . (Hint: for each and there exists a constant such that for all .)
  3. .
  4. , .
  5. .

Exercise 6. If is a linear transformation on a finite-dimensional unitary space, then a necessary and sufficient condition that is that for .

Exercise 7. 

  1. If is a positive linear transformation on a finite-dimensional inner product space, and if is self-adjoint, then for every vector .
  2. Does the conclusion of (a) remain true if is replaced by ?
Bounds of a self-adjoint transformationConvergence of linear transformations
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