Minimax principle

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

Theorem 1. Let $A$ be a self-adjoint transformation on an $n$ -dimensional inner product space $𝒱$ , and let $λ_{1}, \dots, λ_{n}$ be the (not necessarily distinct) proper values of $A$ , with the notation so chosen that $λ_{1} \geq λ_{2} \geq \dots \geq λ_{n}$ . If, for each subspace $ℳ$ of $𝒱$ , $μ (ℳ) = sup {(A x, x) : x in ℳ, ‖ x ‖ = 1},$ and if, for $k = 1, \dots, n$ , $μ_{k} = inf {μ (ℳ) : \dim ℳ = n - k + 1},$ then $μ_{k} = λ_{k}$ for $k = 1, \dots, n$ .

Proof. Let ${x_{1}, \dots, x_{n}}$ be an orthonormal basis in $𝒱$ for which $A x_{i} = λ_{i} x_{i}$ , $i = 1, \dots, n$ ( Section: Spectral theorem ); let $ℳ_{k}$ be the subspace spanned by $x_{1}, \dots, x_{k}$ , for $k = 1, \dots, n$ . Since the dimension of $ℳ_{k}$ is $k$ , the subspace $ℳ_{k}$ cannot be disjoint from any $(n - k + 1)$ -dimensional subspace $ℳ$ in $𝒱$ ; if $ℳ$ is any such subspace, we may find a vector $x$ belonging to both $ℳ_{k}$ and $ℳ$ and such that $‖ x ‖ = 1$ . For this $x = \sum_{i = 1}^{k} ξ_{i} x_{i}$ we have \begin{align} (A x, x) &= \sum_{i=1}^{k} \lambda_{i}|\xi_{i}|^{2}\\ &\geq \lambda_{k} \sum_{i=1}^{k}|\xi_{i}|^{2} \\ &= \lambda_{k}\|x\|^{2}\\ &= \lambda_{k}, \end{align}so that $μ (ℳ) \geq λ_{k}$ .

If, on the other hand, we consider the particular $(n - k + 1)$ -dimensional subspace $ℳ_{0}$ spanned by $x_{k}, x_{k + 1}, \dots, x_{n}$ , then, for each $x = \sum_{i = k}^{n} ξ_{i} x_{i}$ in this subspace, we have (assuming $‖ x ‖ = 1$ ) \begin{align} (A x, x) &= \sum_{i=k}^{n} \lambda_{i}|\xi_{i}|^{2}\\ &\leq \lambda_{k} \sum_{i=k}^{n}|\xi_{i}|^{2} \\ &= \lambda_{k}\|x\|^{2}\\ &= \lambda_{k}, \end{align}so that $μ (ℳ_{0}) \leq λ_{k}$ .

In other words, as $ℳ$ runs over all $(n - k + 1)$ -dimensional subspaces, $μ (ℳ)$ is always $\geq λ_{k}$ , and is at least once $\leq λ_{k}$ ; this shows that $μ_{k} = λ_{k}$ , as was to be proved. ◻

In particular for $k = 1$ we see (using Section: Bounds of a self-adjoint transformation ) that if $A$ is self-adjoint, then $‖ A ‖$ is equal to the maximum of the absolute values of the proper values of $A$ .

EXERCISES

Exercise 1. If $λ$ is a proper value of a linear transformation $A$ on a finite-dimensional inner product space, then $| λ | \leq ‖ A ‖$ .

Exercise 2. If $A$ and $B$ are linear transformations on a finite-dimensional unitary space, and if $C = A B - B A$ , then $‖ 1 - C ‖ \geq 1$ . (Hint: consider the proper values of $C$ .)

Exercise 3. If $A$ and $B$ are linear transformations on a finite-dimensional unitary space, if $C = A B - B A$ , and if $C$ commutes with $A$ , then $C$ is not invertible. (Hint: if $C$ is invertible, then $2 ‖ B ‖ \cdot ‖ A ‖ \cdot ‖ A^{k - 1} ‖ \geq k ‖ A^{k - 1} ‖ / ‖ C^{- 1} ‖$ .)

Exercise 4.

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

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

If $f (λ) = ((1 - λ A)^{- 1} x, y)$ , then $f$ is an analytic function of $λ$ in the region determined by $| λ | < \frac{1}{r (A)}$ (for each fixed $x$ and $y$ ).
There exists a constant $K$ such that $| λ |^{n} ‖ A^{n} ‖ \leq K$ whenever $| λ | < \frac{1}{r (A)}$ and $n = 0, 1, 2, \dots$ . (Hint: for each $x$ and $y$ there exists a constant $K$ such that $| λ^{n} (A^{n} x, y) | \leq K$ for all $n$ .)
$lim sup_{n} ‖ A^{n} ‖^{1 / n} \leq r (A)$ .
$(r (A))^{n} \leq r (A^{n})$ , $n = 0, 1, 2, \dots$ .
$r (A) = lim_{n} ‖ A^{n} ‖^{1 / n}$ .

Exercise 6. If $A$ is a linear transformation on a finite-dimensional unitary space, then a necessary and sufficient condition that $r (A) = ‖ A ‖$ is that $‖ A^{n} ‖ = ‖ A ‖^{n}$ for $n = 0, 1, 2, \dots$ .

Exercise 7.

If $A$ is a positive linear transformation on a finite-dimensional inner product space, and if $A B$ is self-adjoint, then $| (A B x, x) | \leq ‖ B ‖ \cdot (A x, x)$ for every vector $x$ .
Does the conclusion of (a) remain true if $‖ B ‖$ is replaced by $r (B)$ ?

EXERCISES

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