Consider an invertible n ×n matrix A. Can you write A as A = LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of .
Given that A is an n ×n invertible matrix. We claimed that there exists lower triangular matrix L and an orthogonal matrix Q such that A=LQ.
Since and therefore, is also invertible. Now we have PR factorization for ,i.e.
There exists upper triangular matrix R and an orthogonal matrix P such that
By observing in the above factorization. Since R is an upper triangular matrix therefore, will be lower triangular matrix. Also, the matrix is orthogonal and hence it has a factorization for A as lower triangular matrix and an orthonormal matrix.
94% of StudySmarter users get better grades.Sign up for free