How do you find the QR decomposition of a matrix?

The QR decomposition can be extended to rectangular matrices of size m x n , where m ≥ n . In such a case, we have A = QR where Q is of size m x n and its n columns are orthogonal vectors of unit length, and R is an upper triangular matrix of size n x n .

Do all matrices have QR decomposition?

Theorem. Every matrix has a QR-decomposition, though R may not always be invertible.

What is the computational complexity of QR decomposition?

Each iteration step requires the computation of the QR factorization of a full n × n matrix, i.e., each single iteration step has a complexity O(n3).

What is the purpose of QR decomposition?

The QR matrix decomposition allows one to express a matrix as a product of two separate matrices, Q, and R. Q in an orthogonal matrix and R is a square upper/right triangular matrix.

How is QR decomposition is done using Gram-Schmidt method?

The Gram-Schmidt process works by finding an orthogonal projection qn for each column vector an and then subtracting its projections onto the previous projections (qj).

Can QR decomposition fail?

QR Decomposition. For an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n orthogonal matrix Q and an n-by-n upper triangular matrix R so that A = Q*R. The QR decompostion always exists, even if the matrix does not have full rank, so the constructor will never fail.

What is Q in QR decomposition?

A QR decomposition of. a real square matrix A is a decomposition of A as. A = QR, where Q is an orthogonal matrix (i.e. QT Q = I) and R is an upper triangular matrix. If A is nonsingular, then this factorization is unique.

Is the QR decomposition unique?

In class we looked at the special case of full rank, n × n matrices, and showed that the QR decomposition is unique up to a factor of a diagonal matrix with entries ±1.

Does QR decomposition always exist?

The QR decompostion always exists, even if the matrix does not have full rank, so the constructor will never fail. The primary use of the QR decomposition is in the least squares solution of nonsquare systems of simultaneous linear equations.

Why is modified Gram-Schmidt better?

Modified Gram-Schmidt performs the very same computational steps as classical Gram-Schmidt. However, it does so in a slightly different order. In classical Gram-Schmidt you compute in each iteration a sum where all previously computed vectors are involved. In the modified version you can correct errors in each step.