Solvability of New Constrained Quaternion Matrix Approximation Problems Based on Core-EP Inverses

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Advances in Applied Clifford Algebras

Solvability of New Constrained Quaternion Matrix Approximation Problems Based on Core-EP Inverses Ivan Kyrchei, Dijana Mosi´c and Predrag S. Stanimirovi´c∗ Communicated by Rafal Ablamowicz Abstract. Based on the properties of the core-EP inverse and its dual, we investigate three variants of a novel quaternion-matrix (Q-matrix) approximation problem in the Frobenius norm: min AXB − CF subject to the constraints imposed to the right column space of A and the left row space of B. Unique solution to the considered Q-matrix problem is expressed in terms of the core inverse of A and/or the dual core-EP inverse of B. Thus, we propose and solve problems which generalize a well-known constrained approximation problem for complex matrices with index one to quaternion matrices with arbitrary index. Determinantal representations for solutions of proposed constrained quaternion matrix approximation problems obtained. An example is given to justify obtained theoretical results. Mathematics Subject Classification. Primary 15A24, Secondary 15A09, 15A15, 15B33. Keywords. Core-EP inversen, Constrained quaternion matrix approximation problem, Determinantal representation, Noncommutative determinant.

1. Preliminaries In the standard way, fields of complex and real numbers, respectively, are marked with C and R. For the underlying four-dimensional quaternion skew field H = {h0 + h1 i + h2 j + h3 k | i2 = j2 = k2 = ijk = −1, h0 , h1 , h2 , h3 ∈ R}, ∗ Corresponding

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I. Kyrchei et al.

Adv. Appl. Clifford Algebras

the set of all m × n matrices over H is designated by Hm×n , while Hm×n is r reserved for the subset of Hm×n with matrices of rank r. The conjugate h and the norm h of h = h0 + h1 i + h2 j + h3 k ∈ H are given by    h = h0 − h1 i − h2 j − h3 k, h = hh = hh = h20 + h21 + h22 + h23 . The notations rk(A), tr(A), and A∗ present the rank, the trace, and the conjugate transpose of A ∈ Hm×n , respectively. If A ∈ Hn×n and A∗ = A, then A is Hermitian. To more knowledge on quaternion matrices we refer the reader to Zhang [63]. The right quaternionic vector space Hr = Hn×1 is defined using adequate H-valued inner product [49] n


x, yr = y 1 x1 + · · · + y n xn for y = (yi )i=1 , x = (xi )i=1 ∈ Hn×1 .  Defined inner product generates the vector norm x = x, xr on Hn×1 and the quaternion matrix Frobenius norm for A = (ail ) ∈ Hm×n by     ∗ 2 AF = tr(A A) = a.l  = ail 2 . l

It follows from

a.l 2 =



a∗l. , a.l  =




ail ail ,


where a∗l. means the lth row of A∗ and a.l is the lth column of A. Due to noncommutativity in the quaternion skew field, for arbitrary A ∈ Hm×n , it has meaning to define the next notions: – right column space of A: Cr (A) = {v ∈ Hm×1 : v = Au, u ∈ Hn×1 }, – left row space of A: Rl (A) = {v ∈ H1×n : v = uA, u ∈ H1×m }, – right null space of A: Nr (A) = {u ∈ Hn×1 : Au = 0}, – left null space of A: Nl (A) = {u ∈ H1×m : uA = 0}. It is well-known that the Moore–Penrose inverse X = A