Question

Suppose \(A = QR\), where \(Q\) is \(m \times n\) and R is \(n \times n\). Showthat if the columns of \(A\) are linearly independent, then \(R\) mustbe invertible.

Short answer

It is verified that if the columns of \(A\) are linearly independent, then \(R\) is invertible.

Step-by-step solution

\(QR\) factorization of a Matrix

A matrix with order \(m \times n\) can be written as the multiplication of an upper triangularmatrix \(R\) and a matrix \(Q\) which is formed by applying the Gram–Schmidt orthogonalizationprocess to the \({\rm{col}}\left( A \right)\).

The matrix \(R\) can be found by the formula \({Q^T}A = R\).

Finding the matrix \(R\)

Let \({\bf{x}}\) be a vector that satisfies \(R{\bf{x}} = 0\).

Pre-multiply on both sides of \(R{\bf{x}} = 0\) by \(Q\) and simplify.

\(\begin{aligned}{}QR{\bf{x}} = Q \cdot 0\\QR{\bf{x}} = 0\end{aligned}\)

And we have,

\(\begin{aligned}{}A = QR\\A{\bf{x}} = QR{\bf{x}}\\A{\bf{x}} = 0\end{aligned}\)

But the columns of \(A\) are linearly independent; hence the vector \({\bf{x}}\) must be a zero vector.

So, the columns of \(R\) is also linearly independent. Hence the rank of the matrix \(R\) is \(n\).

And by inverse matrix theorem, we have \(R\) is an invertible matrix.