How to force eig() to return a triangular matrix

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Stephan 2017-8-19

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评论： Stephan 2017-8-23

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Hello everyone,

the following code gives an example, where eig() should use the cholesky decomposition, i.e. should return a triangular matrix like chol(). However, it does not.

% positive definite matrix
A = [4,2,-1;
   2,5,1;
  -1,1,6];
% Cholesky works and returns triangular matrix
TriangularMatrix = chol(A);
% no triangular matrix
[V,D] = eig(A);

How do I get a decomposition A = VDV^{-1}, where V is the triangular matrix from a cholesky decomposition.

Thanks for any help!

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John D'Errico 2017-8-19

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I'll tell you this much: The decomposition you are looking for does not exist.

If A is SPD, V lower triangular, D is diagonal, then the product

V*D*inv(V)

will not be symmetric. Therefore there is no need to look for a solution. This is because a lower triangular V will not have an inverse that is the transpose of V. That will happen only if V has very special properties that preclude it being lower triangular, UNLESS it is also diagonal and an identity matrix. So any matrix V that satisfies the requirements will produce a very boring solution, with D==A.

So don't waste your time searching for a solution.

Stephan 2017-8-23