I have some trouble to understand this:

According to the factorization QR, Given a matrix $A\in\mathbb{R}^{n\times p}$, with $rank=min(n,p)$, there exists an orthogonal matrix $Q\in\mathbb{R}^{n\times n}$ and an upper triangular matrix $R\in\mathbb{R}^{n\times p}$ such that

$$A=QR$$ The problem is that when I apply the factorization QR to solve the least squares problem. I suppose that $n>p$, then A is full column rank.

But $Q=[q_1,q_2,...,q_p]\in\mathbb{R}^{n\times p}$ and $R\in\mathbb{R}^{p\times p}$. In this case $Q^TQ=I_p$, however according to several definitions an orthongonal matrix is an square matrix. Then this is not a factorization QR.

Another example is provided here: http://www.seas.ucla.edu/~vandenbe/133A/lectures/ls.pdf in the last slide, the matrix Q found, is not orthogonal since is not square.

moreover, according to the definition orthogonal matrix is a matrix $A$ such that $A^TA=AA^T=I$. But in the example given in the document attached that property does not hold.

Some clarifications, please.


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