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On the Wedderburn-Guttman theorem. (English) Zbl 1111.15001
The paper deals with some extension on the Wedderburn-Guttman theorem [cf. J. H. M. Wedderburn, Lectures on matrices (1934; republ. 1964; Zbl 0121.26101); L. Guttman, Psychometrika 9, 1-16 (1944; Zbl 0060.31212)]. This theorem says that if $$A$$ is an $$m \times n$$ matrix of rank $$a$$, $$M$$ and $$N$$ matrices of sizes $$m \times p$$ and $$n \times p$$, respectively, such that $$M^TAN$$ is nonsingular, then $\text{rank}(A-AN(M^TAN)^{-1}M^TA)=a-p,$ where $$p=\text{rank}(AN(M^TAN)^{-1}M^TA)= \text{rank}(M^TAN)$$.
In this paper the authors focus on the case in which $$M^TAN$$ is rectangular or singular. Let $$M$$ and $$N$$ be $$m \times r$$ and $$n \times s$$ matrices, respectively, where $$r$$ is not necessarily equal to $$s$$, or $$\text{rank}(M^TAN)< min \{r,s\}$$. The authors investigate conditions under which the regular inverse $$(M^TAN)^{-1}$$ can be replaced by a $$g$$-inverse $$(M^TAN)^{-}$$ of some kind, thereby extending the Wedderburn-Guttman theorem. Of course, in this case $$\text{rank}(AN(M^TAN)^{-}M^TA)=g$$ may not be equal to $$\text{rank}(M^TAN)=h$$. Here, the authors study necessary and sufficient condicitons in order to $$\text{rank}(A-AN(M^TAN)^{-}M^TA)=a-g$$. The resultant conditions look similar to those arising in seemingly unrelated contexts, namely Cochran’s and related theorems [cf. P. Šemrl, Linear Algebra Appl. 237–238, 477–487 (1996; Zbl 0847.62043), Section IV] on distributions of quadratic forms involving a normal random vector.

##### MSC:
 15A03 Vector spaces, linear dependence, rank, lineability 15A09 Theory of matrix inversion and generalized inverses
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