# 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
vTivj
(T is transpose)
2) given any matrix M, build
MMT
3) any diagonal matrix with positive entries
4) any diagonal dominant matrix with positive entries
5) the product
M D MT
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
MT 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?

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#### fresh_42

Mentor
2018 Award
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

"All the ways to build positive definite matrices"

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