AMBIENT SPACES FOR CONES GENERATED BY MATRICIAL INNER PRODUCTS

Richard D. Hill, Muriel J. Skoug, Steven R. Waters, and Joseph R. Siler

Abstract. This paper provides an expository survey of the characterizations and properties of four ambient spaces for cones generated by matricial inner products. The partition induced on the vector space of linear operators on complex square matrices is also displayed.

1. Introduction. Throughout this paper we will use M_{n} = M_{n}(ℂ) to denote the space of n × n complex-valued matrices. The real space of n × n hermitian matrices will be denoted by H_{n}, and the space of all linear transformations on M_{n}, by L(M_{n}).

A function ⟨⟨·, ·⟩⟩: M_{n} × M_{n} → M_{n} defined by ⟨⟨A, B⟩⟩ = B^{∗}P A + AQB^{∗}, where P and Q are positive semidefinite and at least one is positive definite provides a vectoral (matricial) inner product [1]. Using this function with various values of P and Q in their Cone Generating Theorem has led Siler and Hill to a lattice of cones [10], which may naturally be thought to reside in any of four real vector spaces. These spaces may be viewed as third–stage generalizations of the real numbers and hermitian matrices in the sense of Barker, Hill, and Haertel [2]. This paper provides a survey of these ambient spaces and their properties.

In Section 2 we summarize information about the spaces HP (the hermitian– preserving linear transformations), RP (the real–preserving linear transformations), SA (the self–adjoint linear transformations), and the set GH which is a proper subspace of SA. In Section 3 we examine the partition of L(M_{n}) induced by intersections of HP, RP, SA, GH, and their complements.

For T ∈ L(M_{n}), we will use ⟨T ⟩ ∈ M_{n^{2}} to denote its representation with respect to basis {E_{ij}}_{i,j=1,... ,n} ⊆ M_{n} ordered antilexicographically. For A ∈ M_{n^{2}}, we find it useful to denote its block form as A = (A_{ij}) = (a_{rs}^{ij}) ∈ M_{n}(M_{n}), (i, j, r, s = 1, …, n). While all eight of the matrix reorderings of Oxenrider and Hill [7] can be used in our study, for brevity we will restrict our results to Ψ and Γ, defined by Ψ(A)_{rs}^{ij} = a_{ri}^{sj} and Γ(A)_{rs}^{ij} = a_{js}^{ir}.

2. The Spaces HP, RP, SA, and GH. The real vector space HP has dimension n^{4} with one basis being given by the transformations represented by the set consisting of: the n^{2} matrices E_{ij} ⊗ E_{ij}; the (n^{4} − n^{2})/2 matrices (E_{ij} + E_{kl}) ⊗ (E_{ij} + E_{kl}); and the (n^{4} − n^{2})/2 matrices (E_{ij} + ιE_{kl}) ⊗ (E_{ij} − ιE_{kl}) for i, j, k, l = 1, …, n, where the last two sets are indexed such that (i, j) ≠ (k, l) and exactly one of (E_{ij} + E_{kl}) and (E_{kl} + E_{ij}) is chosen for fixed i, j, k, l, and ι^{2} = −1; cf. [4].

Proofs of the following characterizations of HP can be found in [3, 4, 5, 8].

\underline{Theorem 2.1}. For T ∈ L(M_{n}), the following are equivalent.

1. T ∈ HP.

2. t_{rs}^{ij} = \overline{t_{ij}^{rs}}, where ⟨T ⟩ = ((t_{ij}^{rs})).

3. There exist A_{1}, …, A_{s} ∈ M_{n}, and real numbers δ_{1}, …, δ_{s}, such that ⟨T ⟩ = ∑_{i=1}^{s} δ_{i}A_{i} ⊗ \overline{A_{i}}, where s ≤ n^{2}.

4. There exist A_{1}, …, A_{s} ∈ M_{n}, and ε_{1}, …, ε_{s}, that assume the values ±1, such that ⟨T ⟩ = ∑_{i=1}^{s} ε_{i}A_{i} ⊗ \overline{A_{i}}, where s ≤ n^{2}.

5. There exist A_{1}, …, A_{s} ∈ M_{n}, and (d_{ij}) ∈ H_{s}, such that ⟨T ⟩ = ∑_{i,j=1}^{s} d_{ij}A_{i} ⊗ \overline{A_{j}}, where s ≤ n^{2}.

6. The block matrix (T (E_{ij}))_{1≤i,j≤n} is hermitian in M_{n^{2}}.

7. Γ(⟨T ⟩) is hermitian in M_{n^{2}}.

8. Ψ(⟨T ⟩) is hermitian in M_{n^{2}}.

9. Γ(⟨T ⟩^{tr}) is hermitian in M_{n^{2}}.

10. Ψ(⟨T ⟩^{tr}) is hermitian in M_{n^{2}}.

11. T is skew–hermitian–preserving.

12. T^{∗} is hermitian–preserving.

13. (T (A))^{∗} = T (A^{∗}).

14. There exists U ∈ L(M_{n}) such that T (A) = U(A) + (U(A^{∗}))^{∗} [5].

The real vector space RP also has dimension n^{4} with one basis being given by the transformations represented by the set consisting of the n^{4} matrices E_{ij} ⊗ E_{kl}, i, j, k, l = 1, …, n.

The following characterizations of RP have proofs much like those for HP. In particular, many of them follow immediately from the specialization of κ̂ to the identity map in Theorem 4.1 of [6].

\underline{Theorem 2.2}. For T ∈ L(M_{n}), the following are equivalent.

1. T ∈ RP.

2. t_{rs}^{ij} = \overline{t_{rs}^{ij}}, where ⟨T ⟩ = ((t_{ij}^{rs})); i.e., ⟨T ⟩ ∈ M_{n^{2}}(ℝ).

3. There exist real numbers t_{rs}^{ij}, for all i, j, r, s = 1, …, n, such that ⟨T ⟩ = ∑_{i,j,r,s=1}^{n} t_{rs}^{ij}E_{ij} ⊗ E_{rs}.

4. There exist A_{1}, …, A_{s} ∈ M_{n}(ℝ), and (d_{ij}) ∈ M_{s}(ℝ), such that ⟨T ⟩ = ∑_{i,j=1}^{s} d_{ij}A_{i} ⊗ A_{j}, where s ≤ n^{2}.

5. The block matrix (T (E_{ij}))_{1≤i,j≤n} is real in M_{n^{2}}.

6. Γ(⟨T ⟩) is real in M_{n^{2}}.

7. Ψ(⟨T ⟩) is real in M_{n^{2}}.

8. Γ(⟨T ⟩^{tr}) is real in M_{n^{2}}.

9. Ψ(⟨T ⟩^{tr}) is real in M_{n^{2}}.

10. T is skew–real–preserving; i.e., T (M_{n}(ιℝ)) ⊆ M_{n}(ιℝ).

11. T^{∗} is real–preserving.

12. \overline{(T (A))} = T (\overline{A}).

13. There exist V_{1}, …, V_{s} ∈ M_{n}(ℝ) and W_{1}, …, W_{s} ∈ M_{n}(ℝ) such that T (A) = ∑_{i=1}^{s} V_{i}AW_{i} for all A ∈ M_{n}.

14. There exists U ∈ L(M_{n}) such that T (A) = U(A) + \overline{(U(\overline{A}))} [5].

The real vector space of self–adjoint linear transformations, SA, also has dimension n^{4} with a basis given by the transformations with representations in the subset of M_{n^{2}} consisting of the n^{2} matrices E_{jj}; the (n^{4}−n^{2})/2 matrices E_{jk} +E_{kj}; and the (n^{4} − n^{2})/2 matrices ιE_{jk} − ιE_{kj}; j, k = 1, …, n^{2}.

For completeness we give the following characterizations of SA.

\underline{Theorem 2.3}. For T ∈ L(M_{n}), the following are equivalent.

1. T ∈ SA.

2. If B is any orthonormal basis for M_{n}, then ⟨T ⟩_{B} is hermitian.

3. ⟨T (A), B⟩ = ⟨A, T (B)⟩ for all A, B ∈ M_{n}.

4. ⟨T (A), A⟩ ∈ ℝ for all A ∈ M_{n}.

5. There exist A_{1}, …, A_{s} ∈ H_{n}, and (d_{ij}) ∈ M_{s}(ℝ), such that ⟨T ⟩ = ∑_{i,j=1}^{s} d_{ij}A_{i} ⊗ A_{j}, where s ≤ n^{2}.

6. The block matrix (T (E_{ij}))_{1≤i,j≤n} represents a hermitian–preserving transformation.

7. Γ(⟨T ⟩) represents a hermitian–preserving transformation.

8. Ψ(⟨T ⟩) represents a hermitian–preserving transformation.

9. Γ(⟨T ⟩^{tr}) represents a hermitian–preserving transformation.

10. Ψ(⟨T ⟩^{tr}) represents a hermitian–preserving transformation.

The hermitian matrices can be characterized in terms of inner products as follows: H_{n} = {A ∈ M_{n} | ⟨Ax, x⟩ ∈ ℝ for all x ∈ ℂ^{n}}, where ⟨x, y⟩ = y^{∗}x. Generalizing this to linear transformations on M_{n} leads naturally to the following definition: GH = {T ∈ L(M_{n}) | ⟨⟨T (A), A⟩⟩ ∈ H_{n} for all A ∈ M_{n}}, where ⟨⟨X, Y ⟩⟩ = Y^{∗}X is a matricial inner product [10].

It can easily be verified that GH is a real vector space of dimension n^{2} with a basis given by the transformations represented by the matrices I ⊗ B_{jk}, where {B_{jk} | j, k = 1, …, n} is any basis for H_{n}. (See Theorem 2.4 (6), below.)

Note in the following characterizations taken from [12] that any T ∈ GH is completely determined by its action on I. Also, the equivalence of (2) and (3) holds for any matricial inner product [10].

\underline{Theorem 2.4}. For T ∈ L(M_{n}), the following are equivalent.

1. T ∈ GH.

2. ⟨⟨T (A), A⟩⟩ ∈ H_{n} for all A ∈ M_{n}.

3. ⟨⟨T (A), B⟩⟩ = ⟨⟨A, T (B)⟩⟩ for all A, B ∈ M_{n}.

4. T (I) ∈ H_{n} and T (A) = T (I)A for all A ∈ M_{n}.

5. ⟨T ⟩ = I ⊗ T (I), where T (I) ∈ H_{n}; i.e., ⟨T ⟩ = (T_{ij}) = ((t_{rs}^{ij})), where i, j = 1, …, n, T_{ij} = O, where i ≠ j, T_{jj} = T (I), and for r, s = 1, …, n, t_{rs}^{jj} = \overline{t_{rs}^{jj}}.

6. There exist A_{1}, …, A_{s} ∈ H_{n}, and real numbers δ_{1}, …, δ_{s}, such that ⟨T ⟩ = ∑_{i=1}^{s} δ_{i}I ⊗ A_{i}, where s ≤ n^{2}.

7. There exist A_{1}, …, A_{s} ∈ H_{n}, and ε_{1}, …, ε_{s}, that assume the values ±1, such that ⟨T ⟩ = ∑_{i=1}^{s} ε_{i}I ⊗ A_{i}, where s ≤ n^{2}.

8. Ψ(⟨T ⟩) = S ⊗ vec(T (I)), where S is the 1 × n^{2} matrix with 1 in each of the positions 1, n + 2, 2n + 3, …, n^{2}, and 0 elsewhere, and T (I) ∈ H_{n}.

9. Γ(⟨T ⟩) = S^{tr} ⊗ (vec(T (I))^{tr})^{tr}, where S is the matrix defined in part 8, and T (I) ∈ H_{n}.

10. The block matrix (T (E_{ij})) = ((e_{j})^{tr} ⊗ T (I)^{(i)}), where T (I) ∈ H_{n}, and T (I)^{(i)} represents the ith column of T (I).

11. T^{∗} ∈ GH.

12. There exists H ∈ H_{n} such that T (A) = HA for all A ∈ M_{n}.

It can easily be verified that GH is neither closed under composition nor under ∗–congruence (conjuctivity). For completeness, we list a few more properties from [12].

\underline{Theorem 2.5}. For T ∈ GH:

1. ⟨T ⟩ = I ⊗ T (I).

2. ⟨T ⟩ is unitarily diagonalizable.

3. If the distinct eigenvalues of T (I) are λ_{1}, …, λ_{s}, with algebraic multiplicities p_{1}, …, p_{s}, respectively, and x_{1}, …, x_{n} is a complete set of orthonormal eigenvectors of T (I), then the distinct eigenvalues of ⟨T ⟩ are λ_{1}, …, λ_{s} with algebraic multiplicities np_{1}, …, np_{s}, respectively, and {e_{j} ⊗ x_{k} | j, k = 1, …, n} is a complete set of orthonormal eigenvectors of ⟨T ⟩.

\underline{Theorem 2.6}. If T ∈ GH, and T (I) is nonsingular, then for any A ∈ M_{n},

1. there exists a constant, c, such that det(T (A)) = c det(A) and

2. rank(T (A)) = rank(A).

3. Interrelationships. While there are 16 possible intersections of the four real spaces HP, RP, SA, and GA and their (set) complements in L(M_{n}), only 11 of these intersections are nonempty. The following Venn diagram shows the partition induced by these spaces with the representative transformations from each piece defined below. We assume that n ≥ 2 since for n = 1, HP = RP = SA = GH are all simply the set of real numbers.

\underline{Definitions of Representative Transformations}. T_{1}(A) = A. T_{2}(A) = tr(A)I, where tr(A) denotes the trace of A. T_{3}(A) = E_{11}AE_{12} + E_{21}AE_{11}. T_{4}(A) = ιE_{21}AE_{21} − ιE_{12}AE_{12}. T_{5}(A) = ιE_{22}AE_{11} − ιE_{11}AE_{22}. T_{6}(A) = E_{11}A. T_{7}(A) = E_{12}AE_{11} + E_{21}AE_{11}. T_{8}(A) = E_{11}AE_{11} + 3E_{12}AE_{11}. T_{9}(A) = ιE_{12}A − ιE_{21}A. T_{10}(A) = (2 + ι)E_{12}AE_{11} + (2 − ι)E_{21}AE_{11}. T_{11}(A) = ιA.

\underline{References}

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2. G. P. Barker, R. D. Hill, and R. D. Haertel, “On the Completely Positive and Positive–Semidefinite–Preserving Cones,” Linear Algebra Appl., 56 (1984), 221–229.

3. M. D. Choi, “Completely Positive Linear Maps on Complex Matrices,” Linear Algebra Appl., 10 (1975), 285–290.

4. R. D. Hill, “Linear Transformations Which Preserve Hermitian Matrices,” Linear Algebra Appl., 6 (1973), 257–262.

5. R. D. Hill and J. R. Siler, “On Hermitian–Invertors,” Missouri Journal of Mathematical Sciences, 8 (1996), 14–21.

6. R. D. Hill and S. R. Waters, “On κ–Real and κ–Hermitian Matrices,” Linear Algebra Appl., 169 (1992), 17–29.

7. C. J. Oxenrider and R. D. Hill, “On the Matrix Reorderings Γ and Ψ,” Linear Algebra Appl, 69 (1985), 205–212.

8. J. A. Poluikis and R. D. Hill, “Completely Positive and Hermitian–Preserving Linear Transformations,” Linear Algebra Appl., 35 (1981), 1–10.

9. J. R. Siler and R. D. Hill, “Reflectors on Vector Spaces,” Linear Algebra Appl., 179 (1993), 1–6.

10. J. R. Siler and R. D. Hill, “Matricial Inner Products and Pointed Cones of Hermitian–Preserving Linear Transformations,” Linear Algebra Appl., 257 (1997), 183–191.

11. M. J. Skoug, R. D. Hill, and J. R. Siler, “On a Lattice of Hermitian Preserving Cones,” Linear Algebra Appl., 367 (2003), 291–300.

12. M. J. Skoug and R. D. Hill, “A Cone of Linear Transformations with Hermitian-Like Properties,” Pan American Mathematical Journal, 13 (2003), 57–65.

Mathematics Subject Classification (2000): 15A48, 15A57 Richard D. Hill Department of Mathematics Idaho State University Pocatello, ID 83209 email: hillrich@isu.edu

Muriel J. Skoug Department of Mathematics and Computer Science Nebraska Wesleyan University Lincoln, NE 68504 email: mjs@nebwesleyan.edu

Steven R. Waters Department of Mathematics Pacific Union College Angwin, CA 94508 email: swaters@puc.edu

Joseph R. Siler Departments of Mathematics and Physics Ozarks Technical Community College Springfield, MO 65801 email: joe siler@hotmail.com