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Weyl inequalities bound the eigenvalues of a sum of self-adjoint endomorphisms

Statement

Let $A, B$ be self-adjoint endomorphisms of an $n$ -dimensional real inner product space. Order the eigenvalues of $A$ , $B$ , and $A + B$ by

$α_{1} \geq \dots \geq α_{n}, β_{1} \geq \dots \geq β_{n}, γ_{1} \geq \dots \geq γ_{n} .$

Then, for integers $1 \leq i, j \leq n$ :

If $i + j - 1 \leq n$ , then $γ_{i + j - 1} \leq α_{i} + β_{j}$ .
If $i + j - n \geq 1$ , then $γ_{i + j - n} \geq α_{i} + β_{j}$ .

Facts & Assumptions

Given: Self-adjoint endomorphisms $A$ and $B$ of an $n$ -dimensional real inner product space.

[L1]

Courant-Fischer characterises ordered eigenvalues by min-max formulas (Courant-Fischer min-max principle for self-adjoint endomorphisms on finite-dimensional real inner product spaces).

[L2]

For finite-dimensional subspaces $U$ and $W$ , one has $\dim (U + W) + \dim (U \cap W) = \dim U + \dim W$ (The dimension formula: for finite-dimensional linear subspaces $U$ and $W$ of $V$ , the subspaces $U + W$ and $U \cap W$ are finite-dimensional and $\dim_{F} (U + W) + \dim_{F} (U \cap W) = \dim_{F} U + \dim_{F} W$ ).

Proof

technique · direct

1.1

Apply [L1] separately to $A$ and $B$ . There are subspaces $U_{i}^{-}, V_{j}^{-}$ of dimensions $n - i + 1, n - j + 1$ such that every nonzero vector in $U_{i}^{-}$ has Rayleigh quotient for $A$ at most $α_{i}$ and every nonzero vector in $V_{j}^{-}$ has Rayleigh quotient for $B$ at most $β_{j}$ ; likewise there are subspaces $U_{i}^{+}, V_{j}^{+}$ of dimensions $i, j$ such that every nonzero vector in them has Rayleigh quotient at least $α_{i}, β_{j}$ respectively.

1.2

If $i + j - 1 \leq n$ , then [L2] gives $\dim (U_{i}^{-} \cap V_{j}^{-}) \geq (n - i + 1) + (n - j + 1) - n = n - i - j + 2 = n - (i + j - 1) + 1$ . Every nonzero $x$ in that intersection satisfies $R_{A} (x) \leq α_{i}$ and $R_{B} (x) \leq β_{j}$ , hence $R_{A + B} (x) = R_{A} (x) + R_{B} (x) \leq α_{i} + β_{j}$ . Therefore $γ_{i + j - 1} \leq α_{i} + β_{j}$ by the min-over-subspaces form of [L1] for $A + B$ .

L1L2algebra

2.1

If $i + j - n \geq 1$ , then [L2] gives $\dim (U_{i}^{+} \cap V_{j}^{+}) \geq i + j - n$ . Every nonzero $x$ in that intersection satisfies $R_{A} (x) \geq α_{i}$ and $R_{B} (x) \geq β_{j}$ , hence $R_{A + B} (x) = R_{A} (x) + R_{B} (x) \geq α_{i} + β_{j}$ . Therefore $γ_{i + j - n} \geq α_{i} + β_{j}$ by the max-over-subspaces form of [L1] for $A + B$ .

L1L2algebra∎

Depends on

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Sources

Christoph Helmberg et al., An interlacing property of the signless Laplacian of threshold graphs (standard reference, not scraped)