Often people asks how to obtain a positive definite matrix. I would like to make a list of all possible ways to generate positive definite matrices (I consider only square real matrices here). Please help me to complete it.(adsbygoogle = window.adsbygoogle || []).push({});

Here M is any matrix, P any positive definite matrix and D any diagonal matrix.

1) given a vector v, build a matrix with entries

v^{T}_{i}v_{j}

(T is transpose)

2) given any matrix M, build

MM^{T}

3) any diagonal matrix with positive entries

4) any diagonal dominant matrix with positive entries

5) the product

M D M^{T}

where D is a diagonal matrix with positive entries

6) if P_{1},P_{2}are positive definite matrices then

P_{1}P_{2}

P_{1}+P_{2}

are also positive definite

7) if P is positive definite then also

M P M^{-1}

P^{-1}

are positive definite

9) the matrix with blocks

P_{1}M

M^{T}P_{2}

is positive definite

10) the product

aP

of a positive scalar a times a positive definite matrix P

11) any submatrix formed as the upper left square matrix of a positive definite matrix (principal minor) is also

a positive definite matrix

.....

do you know other ways not trivially reconducible to one of the above?

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# All the ways to build positive definite matrices

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