求与指定论文一致的商向量空间相同基的技术咨询

I'm currently working through Flows, scaling, and the control of moment hierarchies for stochastic chemical reaction networks and need to replicate the basis of the quotient vector space presented in the paper. Here's the context I have so far:

Key Matrices from the Paper #

Ignoring the physical details behind these matrices for now, here are the examples provided:

• Transition matrix A:

  A = \begin{bmatrix}
      -\alpha & \epsilon & 0 \\
      \alpha & -2\epsilon & \beta \\
      0 & \epsilon & -\beta
  \end{bmatrix}
  

• Matrix Y (partial definition):

  Y = \begin{bmatrix} 1 ...
  

(Quick note: My vector formatting isn't standardized right now—I can't get vectors to avoid occupying their own line—but that's a secondary issue I'm setting aside for the moment.)

General Steps to Find the Matching Quotient Space Basis #

Since quotient vector spaces in this stochastic reaction network context typically tie to invariant subspaces or moment closure constraints, here's how to approach replicating the paper's basis:

1. Clarify the Quotient Space Definition: First, confirm exactly which quotient space the paper references. Common setups include:

   • ℝⁿ / ker(Y) (if Y defines linear constraints on moments)
   • ℝⁿ / im(A) (if working with states modulo the generator matrix's image)
   • ℝⁿ / S where S is an invariant subspace of A.

2. Obtain the Full Matrix Y: The truncated Y provided is insufficient—you'll need the complete matrix from the paper, as it likely defines the subspace we're quotienting by.

3. Compute the Target Subspace:

   • If working with ℝⁿ / ker(Y): Solve Yv = 0 to find the kernel (null space) of Y.
   • If working with an invariant subspace S: Identify all vectors v such that Av ∈ S (adjust based on the paper's specific framing).

4. Derive the Quotient Basis:

   • Start with a standard basis for ℝⁿ (e.g., {e₁, e₂, e₃} for the 3D space here).
   • Remove any basis vectors that lie in the subspace S (the one we're quotienting by).
   • The remaining vectors, when projected onto the quotient space, form a valid basis. You can also use Gaussian elimination to find linearly independent vectors whose equivalence classes span the quotient.

5. Align with the Paper's Basis: Bases are not unique, so once you have your basis, check if it's a scalar multiple or permutation of the paper's version. Adjust via linear combinations to match exactly if needed.

Example Context for Matrix A #

For the 3x3 A matrix provided (a Markov process generator, given its row sums are zero), one common quotient space is the space of transient states modulo the steady-state invariant subspace. To find this:

• Calculate the eigenvalues of A (one will be 0, corresponding to the steady state).
• The invariant subspace for the zero eigenvalue is the kernel of A (since A v = 0 for steady-state vectors).
• The quotient basis would then come from the remaining linearly independent vectors not in this kernel.

Without the full Y matrix, this is a general guide—but once you have the complete Y, follow the steps above to get the exact basis from the paper.

内容的提问来源于stack exchange，提问作者Chris