Early morning I woke up and something in my mind made my go over Strang’s videos on Positive Definite matrices. I had used PDM innumerable times in various courses in my MFE, various places. Somehow i forgot everything I had ever known about Positive Definite Matrices.

So, I decided to write this post which will be a quick recap of this concept in the times to come.

Well, what is it ?

Strang defines as Xt A X >0  then A is positive definite. This means that

  • If all the eigen values of A are positive OR

  • If the determinant of the matrix >0

Well, to interpret geometrically, Xt A X is a quadratic equation and when we are saying that it is greater than 0, it means that it can be expressed as a sum of squares quantity. This means that it has a minimum .

For example x^2 + 2xy + y^2 is nothing but [x y] being X, A being [1    1 , 1    1]  the two rows of A matrix. The det = 0, one of the eigen values =0 . Hence this cannot be positive definite but semi positive definite.

A quadratic equation like 2*x^2 + 12*x*y +7*y^2  does not always be greater than 0….It has a saddle point. So, the matrix that emerges out of this equation is

[ 2  6 ; 6  7 ] has a negative determinant, one of eigen values is negative .so this matrix cannot be positive definite

What so special about positive definite matrices ? it means that there is a minimum and it means that the system is kind of stable at that minimum value.
In most of the physical systems, there has to be a matrix which is positive definite because quadratization makes it have a minimum.

I don’t know , may be after reading the above stuff, you might still want to go and watch strang video!