ON HERMITIAN–INVERTORS

Richard D. Hill and Joseph R. Siler

\underline{Abstract}. Specializing an invertor [20] to a linear transformation on M_{n} for which the image of any hermitian matrix is skew–hermitian yields a hermitian– invertor. This paper gives twelve characterizations of hermitian–invertors and lists other basic results for them. It gives a set of unifying results in a reflector setting and concludes with some remarks on ℤ–linear maps in the Djoković setting [7].

1. Preliminaries. We denote the space of m × n complex matrices by M_{m,n}, M_{n} if m = n, with H_{n} and S_{n} representing its subsets of hermitian and skew– hermitian matrices, respectively. The sets of perhermitian and centrohermitian matrices are denoted P H_{n} and CH_{n}, respectively. A linear transformation T: M_{n} → M_{q} is said to be hermitian–preserving [hermitian–inverting] if and only if T (H_{n}) ⊆ H_{q} [T (H_{n}) ⊆ S_{q}]. The sets of all such maps will be denoted HP_{n,q} and HI_{n,q}, respectively, with the subscripts suppressed in context.

Theorems on linear transformations preserving some set of properties abound in the literature. For example, Volume 33, Numbers 1 and 2 of Linear and Multilinear Algebra [17] is a monograph dedicated to preservers. Included in this monograph is an impressively extensive list of previous preserver papers.

Hermitian–preserving linear transformations in particular have been discussed in [5], [6], [9], and [18]. These ideas permeate other matrix theory, e.g., many inertia results may be cast in terms of hermitian–preservers (cf. [2], [3], [4], [8], [9], [15], [19], [22], [23], and [24], as well as the classical results of Lyapunov [14] and Stein [21]).

The invertor concept of Siler and Hill [20] naturally specializes to the conjugate– transpose reflector ∗ on M_{n}, thus yielding the concept of a hermitian–invertor. As natural as this concept is, it seems to be absent from the literature (as well as the corresponding ideas of perhermitian–invertors, centrohermitian–invertors, etc.).

Hermitian–invertors fill a void in the third level of the Barker–Hill–Haertel table [1], which exists since hermitian–preservers are skew–hermitian–preservers and conversely. (A linear transformation on M_{n} can be uniquely represented as a sum of elements from HP and HI.)

Section 2 gives twelve characterizations of hermitian–invertors in forms motivated by corresponding preserver theorems. Section 3 gives some basic invertor results, also in forms related to certain preserver theorems. Section 4 gives a unifying result in a reflector setting, characterizing invertors of H_{n}, P H_{n}, CH_{n}, and M_{n}(ℝ), paired with the corresponding results for preservers. Finally, Section 5 discusses results for ℤ–linear matrix maps (the Djoković setting [7]).

2. Characterization Theorems. In this section we address the problem of characterizing hermitian invertors; i.e., linear transformations T: M_{n} → M_{q} such that T (H) is skew–hermitian for all H ∈ H_{n}. As in [18], if T: M_{n} → M_{q} is linear, we let < T >∈ M_{nq} be the matrix representation of T with respect to the unit matrices E_{jk} (which have a one in the (j, k) position and zeros elsewhere) in M_{n} and M_{q} respectively, ordered antilexicographically.

We utilize the notation of Poluikis and Hill [18] and Oxenrider and Hill [16]. Two bijections are defined from S = {(j, k) | j = 1, …, q; k = 1, …, n} to {1, …, nq} by [j, k] = (j − 1)n + k and < j, k >= (k − 1)q + j. These correspond to the lexicographical ordering [(j, k) < (r, s) if and only if j < r or (j = r and k < s)] and the antilexicographical ordering [(j, k) < (r, s) if and only if k < s or (k = s and j < r)], respectively, on S. We shall use [j, k] and < j, k > for orderings of S for different q and n; the factors q and n in their formulas will be determined by specification of the ranges of j and k.

Oxenrider and Hill [16] have studied eight element reorderings of matrices in M_{n}(M_{q}) which naturally arise from rearranging the rows or columns, lexicographically or antilexicographically, into n × n blocks ordered lexicographically or antilexicographically. Four of these, which have been used to characterize hermitian– preserving [18], perhermitian–preserving [12], and centrohermitian–preserving [11] linear transformations, are defined by:

\begin{matrix}Γ(T)_{rs}^{jk} = t_{[j,k],[r,s]} = t_{ks}^{jr}, & Ω(T)_{rs}^{jk} = t_{[r,s],[j,k]} = t_{sk}^{rj}, \\ Θ(T)_{rs}^{jk} = t_{<j,k>,<r,s>} = t_{jr}^{ks}, & Ψ(T)_{<r,s>,<j,k>} = t_{rj}^{sk}. \end{matrix}

The following results parallel a number of characterizations of hermitian– preservers found in [5], [6], [9], and [18], of perhermitian–preservers found in [12], of centrohermitian–preservers found in [11], and κ–real and κ–hermitian preservers found in [13].

\underline{Theorem 2.1}. Let T: M_{n} → M_{q} be a linear transformation. Then the following are equivalent:

(1) T is a hermitian–invertor.

(2) T is a skew–hermitian–invertor.

(3) There exist A_{1}, …, A_{s} ∈ M_{q,n} and pure imaginary numbers γ_{1}, …, γ_{s} such that T (X) = ∑_{j=1}^{s} γ_{j}A_{j}XA_{j}^{∗}.

(4) There exist A_{1}, …, A_{s} ∈ M_{q,n} and η_{1}, …, η_{s} ∈ {±i} such that T (X) = ∑_{j=1}^{s} η_{j}A_{j}XA_{j}^{∗}.

(5) There exist A_{1}, …, A_{s} ∈ M_{q,n} and skew–hermitian (g_{jk}) ∈ M_{s} such that T (X) = ∑_{j,k=1}^{s} g_{jk}A_{j}XA_{k}^{∗}.

(6) t_{rs}^{jk} = −\overline{t_{jk}^{rs}} for all (j, k), (r, s) ∈ S, where < T >= ((t_{rs}^{jk})).

(7) The block matrix (T (E_{jk}))_{1≤j≤n; 1≤k≤n} is skew–hermitian.

(8) Γ(< T >) is skew–hermitian.

(9) Ψ(< T >) is skew–hermitian.

(10) Ω(< T >)[= Γ(< T >^{tr})] is skew–hermitian.

(11) Θ(< T >)[= Ψ(< T >^{tr})] is skew–hermitian.

(12) T^{∗} is hermitian–inverting.

\underline{Proof}. Theorem 1.1 of [20] gives us (1) ⇔ (2). Using a proof technique analogous to that of Theorem 1 of [9] (viz., by computing the images of T on the standard basis {E_{jk} + E_{kj}, i(E_{jk} − E_{kj}), E_{jj}} of H_{n} and requiring the images to be skew–hermitian, then assuming (6) and showing these images to be skew–hermitian), we get (1) ⇔ (6).

Assuming (4), a short calculation gives (1). Assuming (1), a proof analogous to that of Theorem 2 of [9] yields (4).

Upon absorbing \sqrt{|γ_{j}|} in A_{j}, (j = 1, …, s), (3) is a restatement of (4). Further, that (5) ⇒ (1) and (3) ⇒ (5) are immediate.

By Lemma 2 of [18], Ψ(< T >) = (T (E_{jk})); this yields (7) ⇔ (9). Since (t_{rs}^{jk}) is skew–hermitian if and only if t_{rs}^{jk} = −\overline{t_{sr}^{kj}}, we have (6) ⇔ (8) ⇔ (9).

Since a matrix is skew–hermitian if and only if its transpose is, we have (8) ⇔ (10) and (9) ⇔ (11).

Finally, since {E_{jk}} is an orthonormal basis for M_{n}, the matrix representation for the Hilbert adjoint of T is < T^{∗} >=< T >^{∗}; this yields (1) ⇔ (12).

In the above we have given summaries of the proofs of the twelve HI characterizations for insight into relationships among them and the corresponding HP theorems. We note that we could get shorter proofs by exploiting the HP = iHI relationship.

Analogous theorems for perhermitian–invertors, centrohermitian invertors, real–invertors, κ–hermitian–invertors, and κ–real–invertors can be given.

3. Some Basic Results. The previously–mentioned “third–stage generalization” of Barker, Hill, and Haertel (Theorem 2 of [1]) gives us that every linear transformation on matrices can be uniquely expressed as the sum of a hermitian– preserver and a hermitian–invertor; viz.

\underline{Theorem 3.1}. If T ∈ L(M_{n}, M_{q}), then there exist unique linear functions H ∈ HP_{n,q} and K ∈ HI_{n,q} such that T = H + K.

This result turns out to be a specialization of Theorem 2.2 of [20]. Analogous results include Theorem 2.11 of [13], Theorem 2.11 of [11], Theorem 2.11 of [12], the Toeplitz (cartesian) decomposition [15], and the Hadamard decomposition A = R_{1} + iR_{2} with R_{1}, R_{2} ∈ M_{n}(ℝ).

Further, since a linear transformation is hermitian–preserving if and only if it is skew–hermitian–preserving, this theorem fills a void in an extension of the Barker– Hill–Haertel table in the third–stage generalization of {pure imaginary number} → {skew–hermitian matrices} → (?).

Results from Hill [9] and Djoković [7] motivate the following three results. Their proofs are similar to the corresponding results for hermitian–preservers in [9].

\underline{Theorem 3.2}. If T ∈ HI, then T (X^{∗}) = −(T (X))^{∗} for all X ∈ M_{n}.

\underline{Theorem 3.3}. If λ is an eigenvalue of algebraic [geometric] multiplicity r of T ∈ HI, then −\overline{λ} is also an eigenvalue of algebraic [geometric] multiplicity r of T.

\underline{Theorem 3.4}. If T ∈ HI_{n}, then the odd elementary symmetric functions E_{1}, E_{3}, … of (the eigenvalues of) T are pure imaginary, whereas the evens E_{2}, E_{4}, … are real. In particular, tr(T) is pure imaginary and det(T) is pure imaginary [real] if n is odd [even].

Analogous to the hermitian–preserving case [Theorem 4 of 18], ∑_{j,k=1}^{s} g_{jk}A_{j} ⊗ \overline{A_{k}} can represent a hermitian–invertor without (g_{jk}) being skew–hermitian. Again linear independence of A_{1}, …, A_{s} is sufficient. The results can be stated formally as follows:

\underline{Theorem 3.5}. Let A_{1}, …, A_{s} be linearly independent in M_{q,n}. Then

(1) ∑_{j,k=1}^{s} g_{jk}A_{j} ⊗ \overline{A_{k}} represents a hermitian–invertor if and only if (g_{jk}) is skew– hermitian.

(2) ∑_{j=1}^{s} γ_{j}A_{j} ⊗ \overline{A_{j}} represents a hermitian–invertor if and only if γ_{1}, …, γ_{s} are pure imaginary.

4. Characterizations in a Reflector Setting. As in [20], a reflector is an additive, involutory map. Any reflector σ on a vector space over a field of characteristic not equal to 2 induces the two additive, idempotent maps spec_{σ}(x) := \frac{1}{2}(x + σ(x)) and skew_{σ}(x) := \frac{1}{2}(x − σ(x)). The ranges of spec_{σ} and skew_{σ} consist of the elements fixed or negated, respectively, by σ.

Let σ be a conjugate–homogeneous reflector on M_{n} and let τ be defined on L(M_{n}) by (τ(T))(A) := σ(T (σ(A))). Then τ is a conjugate–homogeneous reflector on L(M_{n}). By Theorem 3.1 of [20], spec_{τ} L(M_{n}) and skew_{τ} L(M_{n}) consist of the linear preservers and invertors, respectively, of spec_{σ}M_{n} (equivalently, of skew_{σ}M_{n}). Further, by [20, p. 3], skew_{τ} L(M_{n}) = i spec_{τ} L(M_{n}).

These observations, together with the facts that the reflectors σ(A) := \overline{A}, A^{∗}, JA^{∗}J, and J\overline{A}J are conjugate–homogeneous, where J = (δ_{j,n−k+1}), give us the following result:

\underline{Theorem 4.1}. A linear transformation T on M_{n} inverts

(1) H_{n} (and skew–hermitian matrices)

(2) P H_{n} (and skew–perhermitian matrices)

(3) CH_{n} (and skew–centrohermitian matrices)

(4) M_{n}(ℝ) (and M_{n}(iℝ)), respectively, if and only if there exists U ∈ L(M_{n}) such that

(1) T (X) = U(X) − (U(X^{∗}))^{∗}

(2) T (X) = U(X) − J(U(JX^{∗}J))^{∗}J

(3) T (X) = U(X) − J\overline{U(J\overline{X}J)}J

(4) T (X) = U(X) − \overline{U(\overline{X})}.

We now have a first in that this “invertor theorem” motivates a corresponding “preserver theorem.”

\underline{Theorem 4.2}. A linear transformation T on M_{n} preserves (1) H_{n} (and skew–hermitian matrices)

(2) P H_{n} (and skew–perhermitian matrices)

(3) CH_{n} (and skew–centrohermitian matrices)

(4) M_{n}(ℝ) (and M_{n}(iℝ)), respectively, if and only if there exists U ∈ L(M_{n}) such that

(1) T (X) = U(X) + (U(X^{∗}))^{∗}

(2) T (X) = U(X) + J(U(JX^{∗}J))^{∗}J

(3) T (X) = U(X) + J\overline{U(J\overline{X}J)}J

(4) T (X) = U(X) + \overline{U(\overline{X})}.

5. Characterizations in the Djoković Setting.

The setting of this section is as in Djoković [7]. Let ℤ be the center of a division ring D. Assume that D has finite dimension over ℤ and that char D ≠ 2. Let J denote an additive, involutory, reverse–multiplicative map on D.

Define ∗ on M_{n,q} = M_{n,q}(D) by A^{∗} := (b_{jk}), where b_{jk} = a_{kj}^{J}. Then ∗ gives an additive, involutory, reverse–multiplicative map on M_{n,q}.

A matrix A ∈ M_{n} is said to be hermitian [skew–hermitian] if and only if A = A^{∗} [A = −A^{∗}]; a ℤ–linear map T: M_{n} → M_{q} is said to be hermitian–preserving [hermitian–inverting] if and only if T (A) is hermitian [skew–hermitian] whenever A is hermitian.

Djoković’s Theorem (ii) may be restated by renaming his A_{1}, …, A_{p}, B_{1}, …, B_{p} as A_{1}, …, A_{2p} as follows:

\underline{Theorem 5.1}. A ℤ–linear map T: M_{n} → M_{q} is hermitian–inverting if and only if there exist A_{1}, …, A_{2p} ∈ M_{n,q} such that T (X) = ∑_{j,k=1}^{2p} g_{jk}A_{j}^{∗}XA_{k}, where (g_{jk}) = perdiag{1, 1, …, 1, −1, −1, …, −1}, where 1 and -1 each appear with multiplicity p.

Since this (g_{jk}) is skew–hermitian and since its eigenvalues are ±i (each with multiplicity p), we have characterizations in this setting analogous to (3), (4), and (5) of Theorem 2.1.

Further, since this (g_{jk}) is perhermitian we also have characterizations analogous to (3), (4), and (5) of Theorem 4.1 of [12].

\underline{References}

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Richard D. Hill Department of Mathematics Idaho State University Pocatello, ID 83209

Joseph R. Siler Department of Mathematics Ozarks Technical College Springfield, MO 65803

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