a. Find all n × n matrices that are both orthogonal and upper triangular, with positive diagonal entries.
b. Show that the QR factorization of an invertible n × n matrix is unique. Hint: If, then the matrix is both orthogonal and upper triangular, with positive diagonal entries.
If A is an orthogonal and upper triangular matrix, then is lower and upper triangular because and it is an inverse of an upper triangular matrix. Thus,
Because A is orthogonal and diagonal with positive entries must be. So, .
Consider the theorem below.
Products and inverses of orthogonal matrices.
a. The product AB of two orthogonal n × n matrices A and B is orthogonal.
b. The inverse of an orthogonal n × n matrix A is orthogonal.
Thus, according to the above theorem is orthogonal and is upper triangular with positive diagonal entries.
From the part (a) it has
So that, as claimed.
94% of StudySmarter users get better grades.Sign up for free