Min-max theorem#Cauchy interlacing theorem

Let {{mvar|A}} be a {{math|n × n}} Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient {{math|R_A : Cⁿ \ {0} → R}} defined by

:$R\_A(x)\; =\; \backslash frac\{(Ax,\; x)\}\{(x,x)\}$

where {{math|(⋅, ⋅)}} denotes the Euclidean inner product on {{math|Cⁿ}}.

Equivalently, the Rayleigh–Ritz quotient can be replaced by

:$f(x)\; =\; (Ax,\; x),\; \backslash ;\; \backslash |x\backslash |\; =\; 1.$

The Rayleigh quotient of an eigenvector $v$ is its associated eigenvalue $\backslash lambda$ because $R\_A(v)\; =\; (\backslash lambda\; x,\; x)/(x,\; x)\; =\; \backslash lambda$.

For a Hermitian matrix A, the range of the continuous functions R_A(x) and f(x) is a compact interval [a, b] of the real line. The maximum b and the minimum a are the largest and smallest eigenvalue of A, respectively. The min-max theorem is a refinement of this fact.

Let $A$ be Hermitian on an inner product space $V$ with dimension $n$, with spectrum ordered in descending order $\backslash lambda\_1\; \backslash geq\; ...\; \backslash geq\; \backslash lambda\_n$.

Let $v\_1,\; ...,\; v\_n$ be the corresponding unit-length orthogonal eigenvectors.

Reverse the spectrum ordering, so that $\backslash xi\_1\; =\; \backslash lambda\_n,\; ...,\; \backslash xi\_n\; =\; \backslash lambda\_1$.

{{Math theorem

| name = (Poincaré’s inequality)

| note =

| math_statement = Let $M$ be a subspace of $V$ with dimension $k$, then there exists unit vectors $x,\; y\backslash in\; M$, such that

$\backslash langle\; x,\; Ax\backslash rangle\backslash leq\; \backslash lambda\_k$, and $\backslash langle\; y,\; Ay\backslash rangle\; \backslash geq\; \backslash xi\_k$.

}}

{{Math proof|title=Proof|proof=

Part 2 is a corollary, using $-A$.

$M$ is a $k$ dimensional subspace, so if we pick any list of $n-k+1$ vectors, their span $N\; :=\; span(v\_k,\; ...\; v\_n)$ must intersect $M$ on at least a single line.

Take unit $x\; \backslash in\; M\backslash cap\; N$. That’s what we need.

: $x\; =\; \backslash sum\_\{i=k\}^n\; a\_i\; v\_i$, since $x\backslash in\; N$.

: Since $\backslash sum\_\{i=k\}^n\; |a\_i|^2\; =\; 1$, we find $\backslash langle\; x,Ax\; \backslash rangle\; =\; \backslash sum\_\{i=k\}^n\; |a\_i|^2\backslash lambda\_i\; \backslash leq\; \backslash lambda\_k$.

}}

{{Math theorem

| name = min-max theorem

| note =

| math_statement = $$\backslash begin\{aligned\}$$

\lambda_k &=\max _{\begin{array}{c} \mathcal{M} \subset V \\ \operatorname{dim}(\mathcal{M})=k \end{array}} \min _{\begin{array}{c} x \in \mathcal{M} \\ \|x\|=1 \end{array}}\langle x, A x\rangle\\

&=\min _{\begin{array}{c} \mathcal{M} \subset V \\ \operatorname{dim}(\mathcal{M})=n-k+1 \end{array}} \max _{\begin{array}{c} x \in \mathcal{M} \\ \|x\|=1 \end{array}}\langle x, A x\rangle \text{. }

\end{aligned}

}}

{{Math proof|title=Proof|proof=

Part 2 is a corollary of part 1, by using $-A$.

By Poincare’s inequality, $\backslash lambda\_k$ is an upper bound to the right side.

By setting $\backslash mathcal\; M\; =\; span(v\_1,\; ...\; v\_k)$, the upper bound is achieved.

}}

Define the partial trace $tr\_V(A)$ to be the trace of projection of $A$ to $V$. It is equal to $\backslash sum\_i\; v\_i^*Av\_i$ given an orthonormal basis of $V$.

{{Math theorem|name=Wielandt minimax formula|note={{Cite book |last=Tao |first=Terence |title=Topics in random matrix theory |date=2012 |publisher=American Mathematical Society |isbn=978-0-8218-7430-1 |series=Graduate studies in mathematics |location=Providence, R.I}}{{Pg|page=44}}|math_statement=

Let

Define the associated Schubert variety $X\backslash left(V\_1,\; \backslash ldots,\; V\_k\backslash right)$ to be the collection of all $k$ dimensional subspaces $W$ such that $\backslash operatorname\{dim\}\backslash left(W\; \backslash cap\; V\_j\backslash right)\; \backslash geq\; j$.

\lambda_{i_1}(A)+\cdots+\lambda_{i_k}(A)=\sup _{V_1, \ldots, V_k} \inf_{W \in X\left(V_1, \ldots, V_k\right)} tr_W(A)

}}

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=Proof}}

{{Math proof|title=Proof|proof=

The $\backslash leq$ case.

Let $V\_\{j\}\; =\; span(e\_1,\; \backslash dots,\; e\_\{i\_j\})$, and any $W\; \backslash in\; X\backslash left(V\_1,\; \backslash ldots,\; V\_k\backslash right)$, it remains to show that $$$$

\lambda_{i_1}(A)+\cdots+\lambda_{i_k}(A) \leq tr_W(A)

To show this, we construct an orthonormal set of vectors $v\_1,\; \backslash dots,\; v\_k$ such that $v\_j\; \backslash in\; V\_j\; \backslash cap\; W$. Then $tr\_W(A)\; \backslash geq\; \backslash sum\_j\; \backslash langle\; v\_j,\; Av\_j\backslash rangle\; \backslash geq\; \backslash lambda\_\{i\_j\}(A)$

Since $dim(V\_1\; \backslash cap\; W)\; \backslash geq\; 1$, we pick any unit $v\_1\; \backslash in\; V\_1\; \backslash cap\; W$. Next, since $dim(V\_2\; \backslash cap\; W)\; \backslash geq\; 2$, we pick any unit $v\_2\; \backslash in\; (V\_2\; \backslash cap\; W)$ that is perpendicular to $v\_1$, and so on.

The $\backslash geq$ case.

For any such sequence of subspaces $V\_i$, we must find some $W\; \backslash in\; X\backslash left(V\_1,\; \backslash ldots,\; V\_k\backslash right)$ such that $$\backslash lambda\_\{i\_1\}(A)+\backslash cdots+\backslash lambda\_\{i\_k\}(A)\; \backslash geq\; tr\_W(A)$$

Now we prove this by induction.

The $n=1$ case is the Courant-Fischer theorem. Assume now $n\; \backslash geq\; 2$.

If $i\_1\; \backslash geq\; 2$, then we can apply induction. Let $E\; =\; span(e\_\{i\_1\},\; \backslash dots,\; e\_n)$. We construct a partial flag within $E$ from the intersection of $E$ with $V\_1,\; \backslash dots,\; V\_k$.

We begin by picking a $(i\_k-(i\_1-1))$-dimensional subspace $W\_k\text{'}\; \backslash subset\; E\; \backslash cap\; V\_\{i\_k\}$, which exists by counting dimensions. This has codimension $(i\_1-1)$ within $V\_\{i\_k\}$.

Then we go down by one space, to pick a $(i\_\{k-1\}\; -\; (i\_1\; -\; 1))$-dimensional subspace $W\_\{k-1\}\text{'}\; \backslash subset\; W\_k\; \backslash cap\; V\_\{i\_\{k-1\}\}$. This still exists. Etc. Now since $dim(E)\; \backslash leq\; n-1$, apply the induction hypothesis, there exists some $W\; \backslash in\; X(W\_1,\; \backslash dots,\; W\_k)$ such that $$\backslash lambda\_\{i\_1\; -\; (i\_1-1)\}(A|E)+\backslash cdots+\backslash lambda\_\{i\_k-\; (i\_1-1)\}(A|E)\; \backslash geq\; tr\_W(A)$$

Now $\backslash lambda\_\{i\_j\; -\; (i\_1-1)\}(A|E)$ is the $(i\_j-(i\_1-1))$-th eigenvalue of $A$ orthogonally projected down to $E$. By Cauchy interlacing theorem, $\backslash lambda\_\{i\_j\; -\; (i\_1-1)\}(A|E)\; \backslash leq\; \backslash lambda\_\{i\_j\}(A)$. Since $X(W\_1,\; \backslash dots,\; W\_k)\backslash subset\; X(V\_1,\; \backslash dots,\; V\_k)$, we’re done.

If $i\_1\; =\; 1$, then we perform a similar construction. Let $E\; =\; span(e\_\{2\},\; \backslash dots,\; e\_n)$. If $V\_k\; \backslash subset\; E$, then we can induct. Otherwise, we construct a partial flag sequence $W\_2,\; \backslash dots,\; W\_k$ By induction, there exists some $W\text{'}\; \backslash in\; X(W\_2,\; \backslash dots,\; W\_k)\backslash subset\; X(V\_2,\; \backslash dots,\; V\_k)$, such that $$\backslash lambda\_\{i\_2-1\}(A|E)+\backslash cdots+\backslash lambda\_\{i\_k-1\}(A|E)\; \backslash geq\; tr\_\{W\text{'}\}(A)$$ thus

$$\backslash lambda\_\{i\_2\}(A)+\backslash cdots+\backslash lambda\_\{i\_k\}(A)\; \backslash geq\; tr\_\{W\text{'}\}(A)$$ And it remains to find some $v$ such that $W\text{'}\; \backslash oplus\; v\; \backslash in\; X(V\_1,\; \backslash dots,\; V\_k)$.

If $V\_1\; \backslash not\backslash subset\; W\text{'}$, then any $v\; \backslash in\; V\_1\; \backslash setminus\; W\text{'}$ would work. Otherwise, if $V\_2\; \backslash not\backslash subset\; W\text{'}$, then any $v\; \backslash in\; V\_2\; \backslash setminus\; W\text{'}$ would work, and so on. If none of these work, then it means $V\_k\; \backslash subset\; E$, contradiction.

}}{{hidden end}}

This has some corollaries:{{Pg|page=44}}

{{Math theorem|name=Extremal partial trace|note=|math_statement=

$$\backslash lambda\_1(A)+\backslash dots+\backslash lambda\_k(A)=\backslash sup\_\{\backslash operatorname\{dim\}(V)=k\; \}tr\_V(A)$$

$$\backslash xi\_1(A)+\backslash dots+\backslash xi\_k(A)=\backslash inf\_\{\backslash operatorname\{dim\}(V)=k\; \}tr\_V(A)$$

}}

{{Math theorem|name=Corollary|note=|math_statement=

The sum $\backslash lambda\_1(A)+\backslash dots+\backslash lambda\_k(A)$ is a convex function, and $\backslash xi\_1(A)+\backslash dots+\backslash xi\_k(A)$ is concave.

(Schur-Horn inequality) $$$$

\xi_1(A)+\dots+\xi_k(A) \leq a_{i_1,i_1} + \dots + a_{i_k,i_k} \leq \lambda_1(A)+\dots+\lambda_k(A)

for any subset of indices.

Equivalently, this states that the diagonal vector of $A$ is majorized by its eigenspectrum.

}}

{{Math theorem|name=Schatten-norm Hölder inequality|note=|math_statement=

Given Hermitian $A,\; B$ and Hölder pair $1/p\; +\; 1/q\; =\; 1$, $$|\backslash operatorname\{tr\}(A\; B)|\; \backslash leq\backslash |A\backslash |\_\{S^p\}\backslash |B\backslash |\_\{S^q\}$$

}}

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=Proof}}

{{Math proof|title=Proof|proof=

WLOG, $B$ is diagonalized, then we need to show $$

|\sum_i B_{ii} A_{ii} | \leq \|A \|_{S^p} \|(B_{ii})\|_{l^q}

By the standard Hölder inequality, it suffices to show $\backslash |(A\_\{ii\})\backslash |\_\{l^p\}\backslash leq\; \backslash |A\; \backslash |\_\{S^p\}$

By the Schur-Horn inequality, the diagonals of $A$ are majorized by the eigenspectrum of $A$, and since the map $f(x\_1,\; \backslash dots,\; x\_n)\; =\; \backslash |x\backslash |\_p$ is symmetric and convex, it is Schur-convex.

}}{{hidden end}}

Let N be the nilpotent matrix

:

Define the Rayleigh quotient $R\_N(x)$ exactly as above in the Hermitian case. Then it is easy to see that the only eigenvalue of N is zero, while the maximum value of the Rayleigh quotient is {{math|{{sfrac|1|2}}}}. That is, the maximum value of the Rayleigh quotient is larger than the maximum eigenvalue.

The singular values {σ_k} of a square matrix M are the square roots of the eigenvalues of M*M (equivalently MM*). An immediate consequence{{Citation needed|reason=claim is unreferenced and maybe suspicious|date=April 2014}} of the first equality in the min-max theorem is:

:$\backslash sigma\_k^\{\backslash downarrow\}\; =\; \backslash max\_\{S:\backslash dim(S)=k\}\; \backslash min\_\{x\; \backslash in\; S,\; \backslash |x\backslash |\; =\; 1\}\; (M^*\; Mx,\; x)^\{\backslash frac\{1\}\{2\}\}=\backslash max\_\{S:\backslash dim(S)=k\}\; \backslash min\_\{x\; \backslash in\; S,\; \backslash |x\backslash |\; =\; 1\}\; \backslash |\; Mx\; \backslash |.$

Similarly,

:$\backslash sigma\_k^\{\backslash downarrow\}\; =\; \backslash min\_\{S:\backslash dim(S)=n-k+1\}\; \backslash max\_\{x\; \backslash in\; S,\; \backslash |x\backslash |\; =\; 1\}\; \backslash |\; Mx\; \backslash |.$

Here $\backslash sigma\_k^\{\backslash downarrow\}$ denotes the k^th entry in the decreasing sequence of the singular values, so that $\backslash sigma\_1^\{\backslash downarrow\}\; \backslash geq\; \backslash sigma\_2^\{\backslash downarrow\}\; \backslash geq\; \backslash cdots$.

{{Main|Poincaré separation theorem}}

Let {{mvar|A}} be a symmetric n × n matrix. The m × m matrix B, where m ≤ n, is called a compression of {{mvar|A}} if there exists an orthogonal projection P onto a subspace of dimension m such that PAP* = B. The Cauchy interlacing theorem states:

:Theorem. If the eigenvalues of {{mvar|A}} are {{math|α₁ ≤ ... ≤ α_n}}, and those of B are {{math|β₁ ≤ ... ≤ β_j ≤ ... ≤ β_m}}, then for all {{math|j ≤ m}},

::$\backslash alpha\_j\; \backslash leq\; \backslash beta\_j\; \backslash leq\; \backslash alpha\_\{n-m+j\}.$

This can be proven using the min-max principle. Let β_i have corresponding eigenvector b_i and S_j be the j dimensional subspace {{math|S_j {{=}} span{b₁, ..., b_j},}} then

:$\backslash beta\_j\; =\; \backslash max\_\{x\; \backslash in\; S\_j,\; \backslash |x\backslash |\; =\; 1\}\; (Bx,\; x)\; =\; \backslash max\_\{x\; \backslash in\; S\_j,\; \backslash |x\backslash |\; =\; 1\}\; (PAP^*x,\; x)\; \backslash geq\; \backslash min\_\{S\_j\}\; \backslash max\_\{x\; \backslash in$

S_j, \|x\| = 1} (A(P^*x), P^*x) = \alpha_j.

According to first part of min-max, {{math|α_j ≤ β_j.}} On the other hand, if we define {{math|S_m−j+1 {{=}} span{b_j, ..., b_m},}} then

:$\backslash beta\_j\; =\; \backslash min\_\{x\; \backslash in\; S\_\{m-j+1\},\; \backslash |x\backslash |\; =\; 1\}\; (Bx,\; x)\; =\; \backslash min\_\{x\; \backslash in\; S\_\{m-j+1\},\; \backslash |x\backslash |\; =\; 1\}\; (PAP^*x,\; x)=\; \backslash min\_\{x\; \backslash in\; S\_\{m-j+1\},\; \backslash |x\backslash |\; =\; 1\}\; (A(P^*x),\; P^*x)\; \backslash leq\; \backslash alpha\_\{n-m+j\},$

where the last inequality is given by the second part of min-max.

When {{math|n − m {{=}} 1}}, we have {{math|α_j ≤ β_j ≤ α_j+1}}, hence the name interlacing theorem.

{{Main|Trace class#Lidskii's theorem}}

{{Math theorem

| name = Lidskii inequality

| note =

| math_statement = If

& \lambda_{i_1}(A+B)+\cdots+\lambda_{i_k}(A+B) \\

& \quad \leq \lambda_{i_1}(A)+\cdots+\lambda_{i_k}(A)+\lambda_1(B)+\cdots+\lambda_k(B)

\end{aligned}

$$\backslash begin\{aligned\}$$

& \lambda_{i_1}(A+B)+\cdots+\lambda_{i_k}(A+B) \\

& \quad \geq \lambda_{i_1}(A)+\cdots+\lambda_{i_k}(A)+\xi_1(B)+\cdots+\xi_k(B)

\end{aligned}

}}

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=Proof}}

{{Math proof|title=Proof|proof=

The second is the negative of the first. The first is by Wielandt minimax.

$$\backslash begin\{aligned\}$$

& \lambda_{i_1}(A+B)+\cdots+\lambda_{i_k}(A+B) \\

=& \sup_{V_1, \dots, V_k} \inf_{W\in X(V_1, \dots, V_k)}(tr_W(A) + tr_W(B)) \\

=& \sup_{V_1, \dots, V_k} ( \inf_{W\in X(V_1, \dots, V_k)} tr_W(A) + tr_W(B)) \\

\leq& \sup_{V_1, \dots, V_k} ( \inf_{W\in X(V_1, \dots, V_k)} tr_W(A) + (\lambda_1(B)+\cdots+\lambda_k(B))) \\

=& \lambda_{i_1}(A)+\cdots+\lambda_{i_k}(A)+\lambda_1(B)+\cdots+\lambda_k(B)

\end{aligned}

}}{{hidden end}}

Note that $\backslash sum\_i\; \backslash lambda\_i(A+B)\; =\; tr(A+B)\; =\; \backslash sum\_i\; \backslash lambda\_i(A)\; +\; \backslash lambda\_i(B)$. In other words, $\backslash lambda(A+B)\; -\; \backslash lambda(A)\; \backslash preceq\; \backslash lambda(B)$ where $\backslash preceq$ means majorization. By the Schur convexity theorem, we then have

{{Math theorem

| name = p-Wielandt-Hoffman inequality

| note =

| math_statement = $\backslash |\backslash lambda(A+B)\; -\; \backslash lambda(A)\backslash |\_\{\backslash ell^p\}\; \backslash leq\; \backslash |B\backslash |\_\{S^p\}$ where $\backslash |\backslash cdot\backslash |\_\{S^p\}$ stands for the p-Schatten norm.

}}

Let {{mvar|A}} be a compact, Hermitian operator on a Hilbert space H. Recall that the spectrum of such an operator (the set of eigenvalues) is a set of real numbers whose only possible cluster point is zero.

It is thus convenient to list the positive eigenvalues of {{mvar|A}} as

:$\backslash cdots\; \backslash le\; \backslash lambda\_k\; \backslash le\; \backslash cdots\; \backslash le\; \backslash lambda\_1,$

where entries are repeated with multiplicity, as in the matrix case. (To emphasize that the sequence is decreasing, we may write $\backslash lambda\_k\; =\; \backslash lambda\_k^\backslash downarrow$.)

When H is infinite-dimensional, the above sequence of eigenvalues is necessarily infinite.

We now apply the same reasoning as in the matrix case. Letting S_k ⊂ H be a k dimensional subspace, we can obtain the following theorem.

:Theorem (Min-Max). Let {{mvar|A}} be a compact, self-adjoint operator on a Hilbert space {{mvar|H}}, whose positive eigenvalues are listed in decreasing order {{math|... ≤ λ_k ≤ ... ≤ λ₁}}. Then:

::$\backslash begin\{align\}$

\max_{S_k} \min_{x \in S_k, \|x\| = 1} (Ax,x) &= \lambda_k ^{\downarrow}, \\

\min_{S_{k-1}} \max_{x \in S_{k-1}^{\perp}, \|x\|=1} (Ax, x) &= \lambda_k^{\downarrow}.

\end{align}

A similar pair of equalities hold for negative eigenvalues.

{{Math proof|drop=hidden|proof=

Let S' be the closure of the linear span $S\text{'}\; =\backslash operatorname\{span\}\backslash \{u\_k,u\_\{k+1\},\backslash ldots\backslash \}$.

The subspace S' has codimension k − 1. By the same dimension count argument as in the matrix case, S' ∩ S_k has positive dimension. So there exists x ∈ S'  ∩ S_k with $\backslash |x\backslash |=1$. Since it is an element of S' , such an x necessarily satisfy

:$(Ax,\; x)\; \backslash le\; \backslash lambda\_k.$

Therefore, for all S_k

:$\backslash inf\_\{x\; \backslash in\; S\_k,\; \backslash |x\backslash |\; =\; 1\}(Ax,x)\; \backslash le\; \backslash lambda\_k$

But {{mvar|A}} is compact, therefore the function f(x) = (Ax, x) is weakly continuous. Furthermore, any bounded set in H is weakly compact. This lets us replace the infimum by minimum:

:$\backslash min\_\{x\; \backslash in\; S\_k,\; \backslash |x\backslash |\; =\; 1\}(Ax,x)\; \backslash le\; \backslash lambda\_k.$

So

:$\backslash sup\_\{S\_k\}\; \backslash min\_\{x\; \backslash in\; S\_k,\; \backslash |x\backslash |\; =\; 1\}(Ax,x)\; \backslash le\; \backslash lambda\_k.$

Because equality is achieved when $S\_k=\backslash operatorname\{span\}\backslash \{u\_1,\backslash ldots,u\_k\backslash \}$,

:$\backslash max\_\{S\_k\}\; \backslash min\_\{x\; \backslash in\; S\_k,\; \backslash |x\backslash |\; =\; 1\}(Ax,x)\; =\; \backslash lambda\_k.$

This is the first part of min-max theorem for compact self-adjoint operators.

Analogously, consider now a {{math|(k − 1)}}-dimensional subspace S_k−1, whose the orthogonal complement is denoted by S_k−1^⊥. If S' = span{u₁...u_k},

:$S\text{'}\; \backslash cap\; S\_\{k-1\}^\{\backslash perp\}\; \backslash ne\; \{0\}.$

So

:$\backslash exists\; x\; \backslash in\; S\_\{k-1\}^\{\backslash perp\}\; \backslash ,\; \backslash |x\backslash |\; =\; 1,\; (Ax,\; x)\; \backslash ge\; \backslash lambda\_k.$

This implies

:$\backslash max\_\{x\; \backslash in\; S\_\{k-1\}^\{\backslash perp\},\; \backslash |x\backslash |\; =\; 1\}\; (Ax,\; x)\; \backslash ge\; \backslash lambda\_k$

where the compactness of A was applied. Index the above by the collection of k-1-dimensional subspaces gives

:$\backslash inf\_\{S\_\{k-1\}\}\; \backslash max\_\{x\; \backslash in\; S\_\{k-1\}^\{\backslash perp\},\; \backslash |x\backslash |=1\}\; (Ax,\; x)\; \backslash ge\; \backslash lambda\_k.$

Pick S_k−1 = span{u₁, ..., u_k−1} and we deduce

:$\backslash min\_\{S\_\{k-1\}\}\; \backslash max\_\{x\; \backslash in\; S\_\{k-1\}^\{\backslash perp\},\; \backslash |x\backslash |=1\}\; (Ax,\; x)\; =\; \backslash lambda\_k.$

}}

The min-max theorem also applies to (possibly unbounded) self-adjoint operators.G. Teschl, Mathematical Methods in Quantum Mechanics (GSM 99) https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf{{cite book |last1=Lieb |last2=Loss |title=Analysis |edition=2nd |series=GSM |volume=14 |location=Providence |publisher=American Mathematical Society |year=2001 |isbn=0-8218-2783-9 }} Recall the essential spectrum is the spectrum without isolated eigenvalues of finite multiplicity.

Sometimes we have some eigenvalues below the essential spectrum, and we would like to approximate the eigenvalues and eigenfunctions.

:Theorem (Min-Max). Let A be self-adjoint, and let $E\_1\backslash le\; E\_2\backslash le\; E\_3\backslash le\backslash cdots$ be the eigenvalues of A below the essential spectrum. Then

$E\_n=\backslash min\_\{\backslash psi\_1,\backslash ldots,\backslash psi\_\{n\}\}\backslash max\backslash \{\backslash langle\backslash psi,A\backslash psi\backslash rangle:\backslash psi\backslash in\backslash operatorname\{span\}(\backslash psi\_1,\backslash ldots,\backslash psi\_\{n\}),\; \backslash ,\; \backslash |\; \backslash psi\; \backslash |\; =\; 1\backslash \}$.

If we only have N eigenvalues and hence run out of eigenvalues, then we let $E\_n:=\backslash inf\backslash sigma\_\{ess\}(A)$ (the bottom of the essential spectrum) for n>N, and the above statement holds after replacing min-max with inf-sup.

:Theorem (Max-Min). Let A be self-adjoint, and let $E\_1\backslash le\; E\_2\backslash le\; E\_3\backslash le\backslash cdots$ be the eigenvalues of A below the essential spectrum. Then

$E\_n=\backslash max\_\{\backslash psi\_1,\backslash ldots,\backslash psi\_\{n-1\}\}\backslash min\backslash \{\backslash langle\backslash psi,A\backslash psi\backslash rangle:\backslash psi\backslash perp\backslash psi\_1,\backslash ldots,\backslash psi\_\{n-1\},\; \backslash ,\; \backslash |\; \backslash psi\; \backslash |\; =\; 1\backslash \}$.

If we only have N eigenvalues and hence run out of eigenvalues, then we let $E\_n:=\backslash inf\backslash sigma\_\{ess\}(A)$ (the bottom of the essential spectrum) for n > N, and the above statement holds after replacing max-min with sup-inf.

The proofs use the following results about self-adjoint operators:

:Theorem. Let A be self-adjoint. Then $(A-E)\backslash ge0$ for $E\backslash in\backslash mathbb\{R\}$ if and only if $\backslash sigma(A)\backslash subseteq[E,\backslash infty)$.{{rp|77}}

:Theorem. If A is self-adjoint, then

$\backslash inf\backslash sigma(A)=\backslash inf\_\{\backslash psi\backslash in\backslash mathfrak\{D\}(A),\backslash |\backslash psi\backslash |=1\}\backslash langle\backslash psi,A\backslash psi\backslash rangle$

and

$\backslash sup\backslash sigma(A)=\backslash sup\_\{\backslash psi\backslash in\backslash mathfrak\{D\}(A),\backslash |\backslash psi\backslash |=1\}\backslash langle\backslash psi,A\backslash psi\backslash rangle$.{{rp|77}}

• Courant minimax principle
• Max–min inequality

{{Reflist}}

• {{cite arXiv| last1=Fisk|first1=Steve|title=A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices |date=2005|eprint=math/0502408 }}
• {{cite journal|last1=Hwang|first1=Suk-Geun|title=Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices |journal=The American Mathematical Monthly | volume=111| date=2004|issue=2 | pages=157–159|doi=10.2307/4145217|jstor=4145217 | url=https://www.jstor.org/stable/4145217}}
• {{cite journal|last1=Kline|first1=Jeffery|title=Bordered Hermitian matrices and sums of the Möbius function|journal=Linear Algebra and Its Applications |volume=588| date=2020| pages=224–237|doi=10.1016/j.laa.2019.12.004|doi-access=free}}
• {{cite book|last1=Reed |first1=Michael |last2=Simon |first2=Barry | title=Methods of Modern Mathematical Physics IV: Analysis of Operators |year=1978 |isbn= 978-0-08-057045-7 | publisher=Academic Press| url=https://www.elsevier.com/books/iv-analysis-of-operators/reed/978-0-08-057045-7}}

{{Functional analysis}}

{{Analysis in topological vector spaces}}

{{Spectral theory}}

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