Pseudo-inverses Without Matrices
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Advances in Applied Clifford Algebras
Pseudo-inverses Without Matrices Jacques Helmstetter∗ Communicated by Rafal Ablamowicz Abstract. This paper explains that the definition of Moore–Penrose inverses in a given algebra A does not at all require any matrix representation of A. The pseudo-inverses y of a given element x of A (such that xyx = x and yxy = y) involve the right ideals complementary to xA and the left ideals complementary to Ax; the Moore–Penrose inverse corresponds to the complementary right and left ideals selected by means of a positive involution on A. This paper is also the occasion to take the stock of several useful concepts: semi-simple rings, involutions of algebras, especially positive involutions. Mathematics Subject Classification. 16D25, 15A66. Keywords. Pseudo-inverses, Moore–Penrose inverses, Involutions.
1. Introduction Here A is an associative and unital ring; A is always assumed not to be reduced to {0} (in other words, 1 = 0). Often, A will be an algebra of finite dimension over the field R of real numbers. The unit element of A is denoted by 1, unless there is already a usual notation for it, for instance 1V if A is the algebra End(V ) of endomorphisms of a vector space V , or 1n if A is a matrix algebra Mat(n, K). If x is an element of the ring A, every element y of A such that xyx = x and yxy = y is called a pseudo-inverse of x. Conversely, x is a pseudoinverse of y. When x is invertible, the equality xyx = x shows that y = x−1 . When x = 0, the equality yxy = y shows that y = 0; more generally, when x belongs to an ideal J of A, the equality yxy = y shows that y ∈ J. Each of the equalities xyx = x or yxy = y proves that xy and yx are idempotents; consequently, if A contains no idempotent other that 0 and 1, only the invertible or null elements admit a pseudo-inverse, and the pseudoinverse is always unique. When A is commutative, the pseudo-inverse, if it ∗ Corresponding
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J. Helmstetter
Adv. Appl. Clifford Algebras
exists, is also always unique (see Theorem 2.2 below). Nevertheless, it often happens that an element admits infinitely many pseudo-inverses. The main theorem in Sect. 3 involves the right and left ideals xA and Ax generated by the given element x, and implies that the pseudo-inverses y of x are in bijection with the pairs (J, J ) where J is a right ideal such that A = xA ⊕ J, and J a left ideal such that A = Ax⊕J ; each pseudo-inverse y gives the right ideal J = (1 − xy)A and the left ideal J = A(1 − yx) (because xA = (xy)A and Ax = A(yx)). When A is an algebra of finite dimension over the field R of real numbers, the presence of a positive involution τ on A ensures that the set of pseudoinverses of a given element x is never empty; moreover, τ allows us to select in this set a unique pseudo-inverse y such that τ (xy) = xy and τ (yx) = yx. Indeed, the positive involution τ turns A into a Euclidean space where the subspace orthogonal to a right (resp. left) ideal is a right (resp. left) ideal too; in this case, the sele
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