An equation involving upper triangular matrices

[ikebukuro]

Dear all,

 

I came across the following equation involving an upper triangular matrx:

 

MR + R'M' = 2I,

 

where M is a given pxp real-valued matrix, R is an unknown pxp real-valued upper triangular matrix with strictly positive entries on the diagonal and I is a pxp identity matrix. The prime (') denotes matrix transpose. I verified directly for the cases that p = 2 and p = 3 that the solution does not exist always and that the closed-form component-wise solutions are rather cumbersome. I should add that the entries of the matrix M are limited by |Mij| <= 1.

 

Does anyone know the conditions under which the above equation admits a solution for R and of any closed-form solution for R ?

 

 

Thank you in advance!