Using For Solution To Linear Inverse Problems
Compared to the direct matrix inverse, inverse solutions using QR decomposition are more numerically stable as evidenced by their reduced condition numbers .
To solve the underdetermined linear problem where the matrix A has dimensions and rank, first find the QR factorization of the transpose of A:, where Q is an orthogonal matrix (i.e. ), and R has a special form: . Here is a square right triangular matrix, and the zero matrix has dimension . After some algebra, it can be shown that the solution to the inverse problem can be expressed as: 
where is found by Gaussian elimination.
To find a solution to the overdetermined problem which minimizes the norm, first find the QR factorization of A: . The solution can then be expressed as, where and are the same as before, but now is a projection matrix that maps a vector in into .
Read more about this topic: QR Decomposition
Famous quotes containing the words solution, inverse and/or problems:
“There is a lot of talk now about metal detectors and gun control. Both are good things. But they are no more a solution than forks and spoons are a solution to world hunger.”
—Anna Quindlen (b. 1953)
“The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
—Aldous Huxley (1894–1963)
“I have a horror of people who speak about the beautiful. What is the beautiful? One must speak of problems in painting!”
—Pablo Picasso (1881–1973)