Journal of Convex Analysis Volume 3 (1996), No. 2, 371–373

Bounds for Anti-Distance^{∗}

T. Ando

Hokusei Gakuen University, Faculty of Economics, Atsubetsu-ku, Sapporo 004, Japan. e-mail: ando@hokusei.ac.jp

Received January 30, 1995 Revised manuscript received July 26, 1995

Given two matrices A, B, we give an upper and a lower bound for the anti–distance sup_{U} ∥A−U^{∗}BU∥_{∞} where U runs over the set of unitary matrices.

Let 𝕄_{n} be the space of n × n complex matrices, and 𝕌_{n} the group of unitary matrices. For each B ∈ 𝕄_{n}, the set {U^{∗}BU : U ∈ 𝕌_{n}} is the unitary similarity orbit through B. Thus sup_{U∈𝕌_{n}} ∥A − U^{∗}BU∥_{∞} is considered as the anti–distance from A to the orbit with respect to the spectral norm ∥ · ∥_{∞}. In this note we give an upper and a lower bound for the anti–distance. Recall that

∥A∥_{∞} ≡ sup_{∥x∥=1}∥Ax∥ = sup_{∥x∥=∥y∥=1}|y^{∗}Ax|,

where y^{∗} is the conjugate transposed (row) vector of y.

Theorem 1.

sup_{U∈𝕌_{n}} ∥A − U^{∗}BU∥_{∞} ≤ \sqrt{2}∥A ⊗ I − I ⊗ B^{T} ∥_{∞}.  (1)

Proof. Since for any U ∈ 𝕌_{n}

∥A ⊗ I − I ⊗ B^{T} ∥_{∞} = ∥A ⊗ I − I ⊗ (U^{∗}BU)^{T} ∥_{∞}

it suffices to prove

∥A − B∥_{∞} ≤ \sqrt{2}∥A ⊗ I − I ⊗ B^{T} ∥_{∞},  (2)

or equivalently for any (column) vectors x, y

|y^{∗}(A − B)x| ≤ \sqrt{2}∥A ⊗ I − I ⊗ B^{T} ∥_{∞} · ∥x∥ · ∥y∥.  (3)

Remark that

y^{∗}(A − B)x = tr((Ax) · y^{∗} − x · (B^{∗}y)^{∗}).

∗ Research supported in part by Grant-in-Aid for Scientific Research

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

372 T. Ando / Bounds for anti-distance

Since the matrix (Ax) · y^{∗} − x · (B^{∗}y)^{∗} is of rank ≤ 2, the number of its non-zero singular values s_{1} ≥ s_{2} ≥ ⋯ ≥ s_{n}(≥ 0) is 2 at most. Therefore

|y^{∗}(A − B)x| ≤ ∥(Ax) · y^{∗} − x · (B^{∗}y)^{∗}∥_{1}
= s_{1} + s_{2}
≤ \sqrt{2}(s_{1}^{2} + s_{2}^{2})^{1/2}
= \sqrt{2}∥(Ax) · y^{∗} − x · (B^{∗}y)^{∗}∥_{2},  (3)

where ∥ · ∥_{1} and ∥ · ∥_{2} denote the trace norm and the Hilbert–Schmidt norm respectively;

∥X∥_{1} ≡ tr(|X|) and ∥X∥_{2} ≡ (tr(|X|^{2}))^{1/2},

with |X|, the positive semidefinite square root of X^{∗}X. Consider the linear map Δ_{A,B} on the Hilbert space (𝕄_{n}, ∥ · ∥_{2}), defined by

Δ_{A,B}(X) = AX − XB.  (4)

Then clearly we have

(Ax) · y^{∗} − x · (B^{∗}y)^{∗} = Δ_{A,B}(x · y^{∗}).  (5)

On the other hand, since ∥A ⊗ I − I ⊗ B^{T} ∥_{∞} coincides with the operator (spectral) norm of Δ_{A,B} on (𝕄 _{n}, ∥ · ∥_{2}), it follows from (3) and (5) that

\begin{aligned}|y^{∗}(A − B)x| & ≤ \sqrt{2}∥A ⊗ I − I ⊗ B^{T} ∥_{∞} · ∥x · y^{∗}∥_{2} \\ & = \sqrt{2}∥A ⊗ I − I ⊗ B^{T} ∥_{∞} · ∥x∥ · ∥y∥. \end{aligned}

The following special case was proved by Omladic and Semrl [2] by a different method.

Corollary 2. If A, B are normal matrices with eigenvalues {α_{1}, ⋯, α_{n}} and {β_{1}, ⋯, β_{n}} respectively, then

sup_{U∈𝕌_{n}} ∥A − U^{∗}BU∥_{∞} ≤ \sqrt{2} max_{i,j}|α_{i} − β_{j}|.  (6)

Proof. Remark that for general A, B, the set of eigenvalues of A ⊗ I − I ⊗ B^{T} coincides with {α_{i} − β_{j} : i, j = 1, 2, ⋯, n}. If A and B are normal, so is A ⊗ I − I ⊗ B^{T}, hence

∥A ⊗ I − I ⊗ B^{T} ∥_{∞} = max_{i,j}|α_{i} − β_{j}|.  (7)

Now the assertion follows from (1) and (7).

Again in view of (7), for normal A, B the inequality

∥A ⊗ I − I ⊗ B^{T} ∥_{∞} ≤ sup_{U∈𝕌_{n}} ∥A − U^{∗}BU∥_{∞}

is quite trivial. Let us show that this holds in general.

Theorem 3.

∥A ⊗ I − I ⊗ B^{T} ∥_{∞} ≤ sup_{U∈𝕌_{n}} ∥A − U^{∗}BU∥_{∞}.  (8)

Proof. Let us consider the operator Δ_{A,B}, defined by (4), on each of the Schatten p-space (𝕄_{n}, ∥ · ∥_{p}) (1 ≤ p ≤ ∞), where

∥X∥_{p} ≡ (tr(|X|^{p}))^{1/p}.

The operator norm of Δ_{A,B} on (𝕄_{n}, ∥ · ∥_{p}) will be denoted by ∥Δ_{A,B}∥_{p→p}. We used already the fact

∥Δ_{A,B}∥_{2→2} = ∥A ⊗ I − I ⊗ B^{T} ∥_{∞}.  (9)

Since (𝕄_{n}, ∥ · ∥_{p})_{1≤p≤∞} is an interpolation scale (see [1] Chap. III), it follows from the general theory that

∥Δ_{A,B}∥_{2→2} ≤ \sqrt{∥Δ_{A,B}∥_{1→1} · ∥Δ_{A,B}∥_{∞→∞}}.  (10)

By (9) and (10), for (8) it suffices to prove that

∥Δ_{A,B}∥_{1→1} = ∥Δ_{A,B}∥_{∞→∞} = sup_{U∈𝕌_{n}} ∥A − U^{∗}BU∥_{∞}.  (11)

First, since the unit ball of (𝕄_{n}, ∥ · ∥_{∞}) is the convex hull of 𝕌_{n},

\begin{aligned}∥Δ_{A,B}∥_{∞→∞} & = sup_{U∈𝕌_{n}} ∥UA − BU∥_{∞} \\ & = sup_{U∈𝕌_{n}} ∥A − U^{∗}BU∥_{∞}. \end{aligned}

Next

sup_{U∈𝕌_{n}} ∥UA − BU∥_{∞} = sup_{∥T ∥_{1}≤1} sup_{U∈𝕌_{n}} |tr(T UA − T BU)|

= sup_{∥T ∥_{1}≤1} sup_{U∈𝕌_{n}} |tr(U · Δ_{A,B}(T))|

= sup_{∥T ∥_{1}≤1} ∥Δ_{A,B}(T)∥_{1} = ∥Δ_{A,B}∥_{1→1}.

These establish (11) and complete the proof.

References

[1] I. Gohberg, M.G. Krein: Introduction to the Theory of Nonselfajoint Operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969.

[2] M. Omladic, P. Semrl: On the distance between normal matrices, Proc. Amer. Math. Soc. 110 (1990) 591–596.

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