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David Handelman
David Handelman
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Well, the Riccati matrix equation and its variants transform (assuming the leading "coefficient" is invertible) to the quadratic $Z^2 + AZ + B = 0$ (where $Z$ is the unknown matrix and $A,B$ are square). There are generically $C(2n,n)$ solutions, although there could be less, or a continuum (but not greater than $C(2n,n)$ and less than $c$); if $A$ and $B$ commute, the generic number is $2^n$.

One reference is,

D Handelman [me], Fixed points of two-sided fractional matrix transformations, Fixed point theory and its applications, 2007, ID41930, doi:10.1155/2007/41930,

which is freely downloadable.

It's mainly concerned with fixed points of the densely-defined transformation (on $n \times n$ matrices) $X \mapsto (I - CXD)^{-1}$ (where $C$ and $D$ are given), which can be transformed into the quadratic above, and variations on it. It turns out there is a natural graph structure on the solutions, which is generically the Johnson graph.

Edit: Oops, on rereading your question, I see that you required the coefficients to be scalars, which is much more tractible. Oh well, never mind. (If you remember Gilda Radner on SNL ....)

Well, the Riccati matrix equation and its variants transform (assuming the leading "coefficient" is invertible) to the quadratic $Z^2 + AZ + B = 0$ (where $Z$ is the unknown matrix and $A,B$ are square). There are generically $C(2n,n)$ solutions, although there could be less, or a continuum (but not greater than $C(2n,n)$ and less than $c$); if $A$ and $B$ commute, the generic number is $2^n$.

One reference is,

D Handelman [me], Fixed points of two-sided fractional matrix transformations, Fixed point theory and its applications, 2007, ID41930, doi:10.1155/2007/41930,

which is freely downloadable.

It's mainly concerned with fixed points of the densely-defined transformation (on $n \times n$ matrices $X \mapsto (I - CXD)^{-1}$ (where $C$ and $D$ are given), which can be transformed into the quadratic above, and variations on it. It turns out there is a natural graph structure on the solutions, which is generically the Johnson graph.

Well, the Riccati matrix equation and its variants transform (assuming the leading "coefficient" is invertible) to the quadratic $Z^2 + AZ + B = 0$ (where $Z$ is the unknown matrix and $A,B$ are square). There are generically $C(2n,n)$ solutions, although there could be less, or a continuum (but not greater than $C(2n,n)$ and less than $c$); if $A$ and $B$ commute, the generic number is $2^n$.

One reference is,

D Handelman [me], Fixed points of two-sided fractional matrix transformations, Fixed point theory and its applications, 2007, ID41930, doi:10.1155/2007/41930,

which is freely downloadable.

It's mainly concerned with fixed points of the densely-defined transformation (on $n \times n$ matrices) $X \mapsto (I - CXD)^{-1}$ (where $C$ and $D$ are given), which can be transformed into the quadratic above, and variations on it. It turns out there is a natural graph structure on the solutions, which is generically the Johnson graph.

Edit: Oops, on rereading your question, I see that you required the coefficients to be scalars, which is much more tractible. Oh well, never mind. (If you remember Gilda Radner on SNL ....)

Source Link
David Handelman
David Handelman
  • 4.7k
  • 2
  • 23
  • 35

Well, the Riccati matrix equation and its variants transform (assuming the leading "coefficient" is invertible) to the quadratic $Z^2 + AZ + B = 0$ (where $Z$ is the unknown matrix and $A,B$ are square). There are generically $C(2n,n)$ solutions, although there could be less, or a continuum (but not greater than $C(2n,n)$ and less than $c$); if $A$ and $B$ commute, the generic number is $2^n$.

One reference is,

D Handelman [me], Fixed points of two-sided fractional matrix transformations, Fixed point theory and its applications, 2007, ID41930, doi:10.1155/2007/41930,

which is freely downloadable.

It's mainly concerned with fixed points of the densely-defined transformation (on $n \times n$ matrices $X \mapsto (I - CXD)^{-1}$ (where $C$ and $D$ are given), which can be transformed into the quadratic above, and variations on it. It turns out there is a natural graph structure on the solutions, which is generically the Johnson graph.