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An example of Gram Schmidt Orthogonalisation:
Find one orthonormal set of R^3 given vectors (1,1,0)^T,(-1,1,1)^T,(1,-1,0)^T
These vectors can be organized in a data matrix A:
We denote in this article the k-th column of a matrix A by cAk. Similarly we shall use rAj to refer to the j-th row of A. The first orthogonal basis is
To calculate the 2nd orthogonal basis, we have
And the corresponding orthogonal basis is
The 3rd intermediate vector
And the 3rd orthogonal basis
From the calculations above, we have
And the corresponding matrix multiplication form is
QR-Decomposition aka QR-factorization definition:
Let A be a real m*n matrix (m>=n). A can be decomposed into the product A=QR where Q (m*n) is orthogonal (Q^TQ=In) and R (n*n) is upper triangular.