QR Decomposition VI

The following equation defines the QR decomposition:

where m is the number of rows and n is the number of columns in A, Q is an m-by-m unitary matrix, R is an m-by-n upper trapezoidal matrix, R₁ is a k-by-k upper triangular matrix where k is the minimum of m and n, R₂ is an m-by-(n-m) submatrix of R, and 0 is an (m-n)-by-n zero matrix.

You can use QR decomposition to calculate the determinant of a square matrix. For example, consider the following equation: det(A) = det(Q)*det(R). Because Q is orthogonal, the following is true: |det(Q)| = 1. Thus, the following also is true:

You also can use QR decomposition to solve the least-squares problem of a linear equation Ax = b when A is full rank and m ≥ n. For example, consider the following equation:

where the following are true:

• 
• 
• The size of Q₁ is m-by-n
• The size of Q₂ is m-by-(m-n)
• The size of R₁ is n-by-n.

Because min(||b – Ax||₂) depends on min(||Q₁^Tb – R₁x||₂), you can obtain the solution x by solving the following new linear equation: R₁x = Q₁^Tb.