linear algebra - Basic concepts about matrices and their decompositions

I am studying the basics of linear algebra and I have some questions that I can not conclude them by my own.

Let $A \in \Bbb R^{m \times n}$

• $A$ can always be expressed as a LU decomposition?


• $A$ can always be expressed in the reduced echelon form?


• $A$ can be expressed as a QR decomposition?

Answer

• First one: by definition a triangular matrix is square (wait, that sounds funny)...so the first one is not really applicable in general to $\mathbb{R}^{m\times n}$ - it is also not true that all square matrices have a LU decomposition. For necessary and sufficient conditions see this article: http://arxiv.org/pdf/math/0506382v1.pdf


• Yes, row equivalence is an equivalence relation, and every equivalence class contains one matrix in reduced echelon form. You can prove it by induction on $m$. As a reference you can refer to Matrices and linear transformations, theorem 1.18 [Cullen].


• Yes, ref: Proposition 16.11 in The linear algebra a beginning graduate student ought to know [Golan]