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.

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 is transpose)

2) given any matrix M, build

MM

3) any diagonal matrix with positive entries

4) any diagonal dominant matrix with positive entries

5) the product

M D M

where D is a diagonal matrix with positive entries

6) if P

P

P

are also positive definite

7) if P is positive definite then also

M P M

P

are positive definite

9) the matrix with blocks

P

M

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?

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 thenP

_{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}MM

^{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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