Matrix Partial Orders Associated with Space Preorder
In this expository article, we discuss some fundamentals of well-known matrix partial orders that are closely associated with space preorder on rectangular matrices. Particularly, we consider partial order defined by space decomposition, star ordering, an
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Abstract In this expository article, we discuss some fundamentals of well-known matrix partial orders that are closely associated with space preorder on rectangular matrices. Particularly, we consider partial order defined by space decomposition, star ordering, and minus partial order for our discussion. These relations are closely associated with comparison of column spaces and row spaces of matrices. Results associated with selected matrix relation that are known in the literature along with some interesting observations are put together. At many places, though the proofs of several results are known in the past literature, by part or completely, for better reading purpose, independent proofs are provided. Keywords Generalized inverse · Partial order · Preorder · Star partial order · Space decomposition · Minus partial order Mathematics Subject Classification (2010) 15A09
1 Preliminaries In this section, we provide preliminaries such as notation, definitions, and basic results required on matrices, generalized inverse, and partial order on a set.
K. Manjunatha Prasad · K.S. Mohana (B) · Y. Santhi Sheela Department of Statistics, Manipal University, Manipal 576104, India e-mail: [email protected] K. Manjunatha Prasad e-mail: [email protected] K. Manjunatha Prasad e-mail: [email protected] Y. Santhi Sheela e-mail: [email protected] K.S. Mohana Department of Mathematics, Manipal Institute of Technology, Manipal 576104, India R.B. Bapat et al. (eds.), Combinatorial Matrix Theory and Generalized Inverses of Matrices, DOI 10.1007/978-81-322-1053-5_17, © Springer India 2013
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1.1 Matrices and Generalized Inverses Let C, R, F denote the complex, real, and arbitrary fields, respectively. We use the notation K for a scalar field when the concerned scalar field can be either C or R by choice. Let Fn denote the vector space of all n-tuples (usually written columnwise) with entries from F. Fm×n denotes the set of all m × n matrices, which is also a vector space over F. The dimension of a vector space V is denoted by D(V ). The class of all matrices defined over F is denoted by Mat(F). Given two vector spaces V and W , a mapping T :V →W is called a linear mapping if T is additive (T (v + w) = T (v) + T (w) ∀ v, w ∈ V ) and preserves scalar multiplication (T (λv) = λT (v) for all scalars λ and v ∈ V ). An additive mapping T on a complex vector space is called conjugate linear if ¯ (v). T (λv) = λT A ∈ Fm×n can also be treated as a linear transformation A : Fn → Fm . The matrix denoted by A (A∗ in case F = C) is an n × m matrix such that the (i, j )th entry of A equals the (j, i)th entry of A (conjugate of the (j, i)th entry in the complex case). The (i, j )th entry of a matrix A is generally denoted by Aij . A is called the transpose of A, and A∗ in the case of complex numbers is called conjugate transpose of A. A complex (real) matrix is said to be Hermitian (symmetric) if A = A∗ (A = A ). The class of all n × n Hermitian matrices is denoted by Hn . The vector space sp
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