All the ways to build positive definite matrices

[lukluk]

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_iv_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 P₁,P₂ are positive definite matrices then
P₁ P₂
P₁ +P₂
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
P₁ M
M^T P₂
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?

 

[fresh_42]

You want to solve the quadratic polynomial equation $\displaystyle{\sum_{i,j}} a_{ij}x_ix_j>0$ for all possible $x_\, , \,\text{not all }x_i=0$. If we let run the $x$ through $(1,0,\ldots,0),\ldots,(0,\ldots,0,1)$ then we get $n$ linear constraints for $n^2$ variables $a_{ij}$. These are really many possible solutions. E.g. any symmetric real square matrix $A$ is positive definite, if $A=GG^\tau$ with a regular lower triangular matrix. See also https://en.wikipedia.org/wiki/Definiteness_of_a_matrix