On the Entries of Orthogonal Projection Matrices
The present paper is concerned with characterizing entries of orthogonal projectors (i.e., a Hermitian idempotent matrices). On the one hand, several bounds for the values of the entries are identified. On the other hand, particular attention is paid to t
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Abstract The present paper is concerned with characterizing entries of orthogonal projectors (i.e., a Hermitian idempotent matrices). On the one hand, several bounds for the values of the entries are identified. On the other hand, particular attention is paid to the question of how an orthogonal projector changes when its entries are modified. The modifications considered are those of a single entry and of an entire row or column. Some applications of the results in the linear regression model are pointed out as well. Keywords Orthogonal projector · Idempotent matrices · Oblique projector · Moore–Penrose inverse · Linear model Mathematics Subject Classification (2010) 15A09 · 62J12
1 Preliminaries Let Cm,n (Rm,n ) denote the set of m × n complex (real) matrices. The symbols M∗ , R(M), N (M), and rk(M) will stand for the conjugate transpose, column space (range), null space, and rank of M ∈ Cm,n , respectively. Moreover, In will be the identity matrix of order n, and for a given M ∈ Cn,n we define M = In − M. Additionally, tr(M) will stand for the trace of M ∈ Cn,n . For two matrices M and N having the same number of rows, the columnwise partitioned matrix obtained by juxtaposing the two matrices will be denoted by (M : N). In Sect. 4, which provides some results dealing with real matrices, we will use the symbol M to denote the transpose of M ∈ Rm,n . O.M. Baksalary (B) Faculty of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Pozna´n, Poland e-mail: [email protected] G. Trenkler Department of Statistics, Dortmund University of Technology, Vogelpothsweg 87, 44221 Dortmund, Germany e-mail: [email protected] R.B. Bapat et al. (eds.), Combinatorial Matrix Theory and Generalized Inverses of Matrices, DOI 10.1007/978-81-322-1053-5_9, © Springer India 2013
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O.M. Baksalary and G. Trenkler
We define two particular vectors in Cn,1 , namely 1n , which denotes n × 1 vector of ones, and ei , which stands for a vector having 1 in the ith row and all other entries equal to zero, i.e., 1n = (1, 1, . . . , 1)∗
and ei = (0, . . . , 0, 1, 0, . . . , 0)∗ ,
i = 1, 2, . . . , n.
Recall that any vectors x, y ∈ Cn,1 satisfy the Cauchy–Schwarz inequality, which reads |x∗ y| ≤ x y. A matrix M ∈ Cn,n satisfying M2 = M is called idempotent. It is known that every idempotent matrix represents an oblique projector onto its column space R(M) along its null space N (M). A key role in the subsequent considerations will be played by a subset of idempotent matrices consisting of matrices which are additionally Hermitian. Such matrices are called orthogonal projectors, and their set will be denoted by COP n , i.e., 2 ∗ COP n = M ∈ Cn,n : M = M = M . ⊥ ⊥ A projector M ∈ COP n projects onto R(M) along R(M) , where R(M) denotes the orthogonal complement of R(M). An important role in considerations dealing with projectors is played by the notion of the Moore–Penrose inverse, for M ∈ Cm,n defined to be the unique matrix M† ∈ Cn,m satisfying the equations: ∗ ∗ MM† M = M, M† MM† = M† , MM† =
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