QR Decomposition - Using For Solution To Linear Inverse Problems

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: 
x = Q
\begin{bmatrix} (R_1^T)^{-1}b \\ 0 \end{bmatrix}
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 .

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