By Jeremy D Finn
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Additional info for A General Model for Multivariate Analysis (International Series in Decision Processes)
S The Algebra of Matrices 31 ~ The_@~ a matrix can never exceed the smaller of its two dimensions. (""'--~----------------~~--------~------------···--·-·--:! _ ----· · --~-~-~-As an example, let A=[~ 22 121 3 0 The rank of A is 3 since no column can be expressed as a linear combination of other columns. ). The matrix is of full rank 2, as long as the second column (or row) is not a linear function of the first. ~.. ~~ei r .. ,~~ .... ~.. _ ~ r(AB),;; min [r(A), r(B)] Further, if A is mxn and B i5'7ix/, and both··areoTr;~k n, then r(AB)=n.
B* is termed an orthonormal basis for the columns of B~ Again, if the rank of B is less than n, one or more of the diagonal elements ofT will be null. These are the normalizing constants (lengths) for the orthogonal vectors. The column of B that is dependent upon preceding columns will go to zero in s·\ and would have zero length. This can occur either by columns of B being direct linear combinations of one another or by B having more columns than rows. ·61esfileniselves. That is b; is the part o(t>;iilat"istridei)endent ----········-··--~~----·-··•:J:''""' .
Schmidtpr6ceau're(l3fork;T967rancfotfi'er'meffioos, such as that of Householder (1964), are more accurate and should be employed for large problems. The Gram-Schmidt method operates by successively computing columns of X that are orthogonal to preceding columns, at each stage normalizing the newly computed vector. Bjork's modification postpones normalization until all orthogonal vectors are computed. It is also possible to produce a factorization which is orthogonal with respect to an mxm diagonal matrix metric D.
A General Model for Multivariate Analysis (International Series in Decision Processes) by Jeremy D Finn