Abstract
Definition 2.1. Let \(a_{ij} \in \mathcal{F},\) i = 1,2,…,m, j = 1,2,…,n, where \(\mathcal{F}\) is a suitable space, such as the one-dimensional Euclidean or complex space.
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Notes
- 1.
In most of the literature, the subscript is typically omitted. In this volume we shall include it more often than not for the greater clarity it brings to the discussion.
- 2.
Strictly speaking, we should also provide an argument based on the rows of A and on BA = I m , but this is repetitious and is omitted for the sake of simplicity.
- 3.
This involves some loss of generality but makes a proof by elementary methods possible. The results stated in the proposition are valid without these restrictive assumptions.
- 4.
This is an assumption that simplifies the proof considerably. Strictly speaking, it is not necessary. The cost it imposes on the generality of the result is, in any event, miniscule in view of Proposition 2.40.
- 5.
A simple proof of this is as follows. Suppose there exists a non-null vector c such that \(X_{j}^{{\prime}}X_{j}c = 0.\) But \(c^{\prime}X_{j}X_{j}c = 0,\) implies X j c = 0, which is a contradiction.
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Dhrymes, P.J. (2013). Matrix Algebra. In: Mathematics for Econometrics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8145-4_2
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