Transformation of inverse to a system of linear equations. Need to solve X=(U′WU)−1U′. U′ is U′ is 7x7 positive definite matrix, U′ is of rank 3

Pavukol 2022-09-11 Answered
Transformation of inverse to a system of linear equations. Need to solve X = ( U W U ) 1 U . U is U is 7 × 7 positive definite matrix, U is of rank 3.
Transformed ( U W U ) 1 U as
( U W U ) 1 U W U = I X W U = I U W X = I ( I U W ) v e c ( X ) = v e c ( I ) .
When I solved X = ( U W U ) 1 U and as the above linear system using R, the answers are slightly different. Does something wrong with the above logic?
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Answers (1)

faliryr
Answered 2022-09-12 Author has 15 answers
1) A standard and the simplest way (if there's nothing better to do) to compute X is first to compute the Cholesky factorization of the SPD matrix U W U = L L ′ and then solve the system with multiple right-hand sides L L X = U . Note that U W U is 3 × 3 so this approach is very cheap.
2) With the approach in question, the problem is already in finding X from X W U = I, which is not uniquely solvable [any 3 × 7 matrix Y whose rows are orthogonal to the columns of W U will also satisfy ( X + Y ) W U = I.
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