Question

Consider an n x m matrix A with rank (A) = m. Is it always possible to write A as A = QL where Q is an n x m matrix with orthonormal columns and L is a lower triangular m x m matrix with positive diagonal entries? Explain.

Short answer

Yes, it is always possible to writeA as QL.

Step-by-step solution

Determine QR factorization of matrix

In QR factorization of a matrix A, the matrix is decomposed into a product A = QR, where Q is an orthonormal matrix and R is an upper triangular matrix.

Represent the diagonal entries

Suppose there is an n x m matrix A, where the rank of matrix A is m.

Now, it is given that rank of A is m, so, A contains m linearly independent columns. This implies that for the column space of A, an orthonormal basis made by m columns of Q.

Now, the matrix A becomes A = QL, where the distinction of basis matrix is L and it is lower triangular matrix with positive diagonal matrix.

The representation is distinctive and the entries are given by:

$l_{11}=||{\stackrel{\rightharpoonup }{v}}_1||,l_{jj}=||{\stackrel{\rightharpoonup }{v}}_2||\left(forj=2,...,m\right),l_{ij}={\stackrel{\rightharpoonup }{u}}_i·{\stackrel{\rightharpoonup }{v}}_j\left(fori>j\right)$

So, the linearly independent columns of A are ${\stackrel{\rightharpoonup }{v}}_1,{\stackrel{\rightharpoonup }{v}}_2,...,{\stackrel{\rightharpoonup }{v}}_m$and the orthonormal columns ${\stackrel{\rightharpoonup }{u}}_1,{\stackrel{\rightharpoonup }{u}}_2,...,{\stackrel{\rightharpoonup }{u}}_m$ are of Q.

Thus, we can always write A = QL.