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I was wondering how I can generate about 50 random 2x2 symmetrical positive definite matrices in PYTHON. Is there a block of code using scikit or numpy anyone can guide me to?

ML Enthusiast
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  • For any matrix A, the matrix A*A is positive semidefinite, https://stackoverflow.com/questions/619335/a-simple-algorithm-for-generating-positive-semidefinite-matrices .. – StupidWolf Nov 18 '20 at 00:02
  • The question is about positive *definite* matrices, though. – Antimon Nov 18 '20 at 00:06
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    A symmetric matrix is positive definite iff all its eigenvalues are positive. Why not make a random diagonal matrix with positive diagonal? – bnaecker Nov 18 '20 at 00:12

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Strictly diagonally dominant matrices are positive definite. So you could generate a random A, compute AA= A'A and then increase the elements on the diagonal to make sure that

AA[i,i] > sum( abs(AA[i,j]), j != i ),

e.g., compute the sum on the right-hand side and then add 1 to it and assign the result to AA[i,i].

More simply, you could compute A'A + alpha*I for some alpha > 0 of your choice and where I is the identity matrix. All eigenvalues of this matrix are >= alpha, which make it "safely" positive definite.

You can also use sklearn to do it easier and then make matrix values random. See: https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_spd_matrix.html

FilipA
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A real symmetric 2-by-2 matrix

/ a b \
\ b c /

is positive definite iff its trace and determinant are positive. As the determinant is ac-b^2 this happens iff a and c are positive and -m < b < m where m is the geometric mean \sqrt ac.

You can therefore draw positive a and c and a factor f with -1<f<1 and then use

/     a         f * sqrt(ac) \
\ f * sqrt(ac)      c        /

Note that this method can easily be vectorized.

Paul Panzer
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"Random" means different things to different people but I'll take a stab.

As a previous answer noted, a real matrix X is positive semi-definite if (and only if) it can be factored as X=M'M, i.e. it is a Gram matrix.

One simple way to make X random, while also guaranteeing positive definiteness, is to use random normal entries for M. See also, the Wishart distribution.

shambo
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