Multidimensional Analogies of One-dimensional concepts

Questions
1

Hello,
I would like to ask are there exist more such analogies like:

  1. matrix can be positive semi-definite - number/function is nonnegative
  2. eigenvalues of system - zeros of function ( I’m not sure if its true..)

I’m trying to as many analogies and corresponding equations as I can to help me with understanding theory of matrices, systems and multidimensional things.

Thanks in advance :slight_smile:

2

the idea of positive semi-definite matrices can be extended to define matrix inequalities

3

There are matrix power series, a matrix exponential (can be defined in terms of power series), and more. But I should point out that all this is somewhat removed from the intent of this forum.

4

How can a matrix inequality be implemented in Mathematica?

5

One way, probably not the best, would be to check the eigenvalues of the matrix difference.

matrixGreaterEqual[m1_, m2_] /; 
  MatrixQ[m1, Element[#, Reals] &] && 
   MatrixQ[m2, Element[#, Reals] &] && 
   Dimensions[m1] == Dimensions[m2] && Apply[Equal, Dimensions[m1]] :=
  Module[{diffmat = m1 - m2, evals},
  evals = Eigenvalues[diffmat];
  VectorQ[evals, Element[#, Reals] &] && Min[evals] >= 0]

Example:

m1 = {{5, -1}, {-1, 3}};
m2 = {{2, -1}, {-1, 2}};

In[11]:= matrixGreaterEqual[N@m1, m2]

(* Out[11]= True *)

There is probably a better way to go about this using CholeskyDecomposition.

6

I’d like to be able to implement it symbolically to do convex relaxation.

7

Is convex relaxation where you take a nap inside an ellipse?

Any symbolic effort in this direction will, at best, only handle toy problems. It’s just hugely worse in terms of computational complexity.

8

What I should have said is that I’d like a set of equations involving the matrix elements that insure it’s positive semi-definite. Sylvester’s criterion is one possibility, I suppose.

9

If you have a starting point that satisfies the LMI constraints, it suffices, in theory, just to keep the determinants positive (a change in sign means an eigenvalue became negative). In practice this could fail if a step size was too large and you jump into an region of the matrix space where, say, two eigenvalues became negative.