To determine all pairs ((\mathcal{S}, \mathcal{I})) where (\mathcal{S}) is a linear subspace of the span (\mathcal{G} = \operatorname{span}{G_1, \ldots, G_n}) and (\mathcal{I}) is a non-trivial invariant subspace for every (S \in \mathcal{S}), follow these steps:

1. Understand Invariant Subspaces

A subspace (\mathcal{I} \subseteq \mathbb{C}^m) is invariant under (S) if ...

Let V = C^m and G = span{G1,…,Gn} ⊆ End(V). For a subspace I ⊆ V and a linear subspace S ⊆ G, “I is invariant under every S ∈ S” means S(I) ⊆ I for all S ∈ S.

Key observation (reduction to a rank condition)

• Fix k with 1 ≤ k ≤ m−1 and a k-dimensional subspace I ⊆ V.
• Let i_I: I → V be the inclusion and q_I: V → V/I the quotient map.
• Define the linear map Φ_I: G → Hom(I, V/I) by Φ_I(A) = q_I ∘...

This is an interesting problem in linear algebra that touches on the theory of invariant subspaces and simultaneous similarity. Let me provide a systematic approach to finding these pairs .

Structure of the Problem

The key insight is that we're looking for pairs where:

• is a subspace of matrices
• $\mathcal{I} \subseteq \mathbb{...

Notation that will be used throughout
V = C^m, M = M m×m (C) ,
G = span{ G1 ,…, Gn } ⊂ M, (1)
A = alg(G1 ,…, Gn ) = the unital algebra generated by G.

For a subspace I ⊂ V denote
L(I) = { X ∈ M : X(I) ⊂ I } (2)

L(I) is a linear subspace (in fact a sub-algebra) of M.
For a fixed I put

Smax(I) = G ∩ L(I). (3)

Obviously

S ⊂ G and X(I...

Introduce the notation

 • 𝐺 = span{G1 ,…, Gn } ⊂ 𝑀m (ℂ).
 • for a subspace I ⊂ ℂm let

  Stab𝐺(I) = { A ∈ 𝐺 | A I ⊂ I } (∗)

  (the stabiliser of I inside 𝐺).

The required pairs (𝒮 , 𝑰) are exactly the pairs

  (𝒮 , 𝑰) with 𝑰 ⊂ ℂm   non–--trivial and 𝒮 ⊂ Stab𝐺(𝑰).

So the whole problem reduces to the following two tasks.

 (1) Find every subspac...

Recall that a subspace I is invariant under a matrix S if S(I) ⊆ I, meaning that for every vector v in I, ...

This is an excellent and deep question that lies at the intersection of linear algebra, representation theory, and computational algebra. The problem of finding all such pairs is complex, but it can be broken down into a structured approach.

Let's rephrase the core problem: For a given subspace of matrices , we are looking for the se...

The short answer is: nothing mystical is hiding here—you have reduced your problem to the study of the module
 ℂ^m
over the (finite‐dimensional) associative algebra
 A := span_{ℂ}{G₁,…,Gₙ}
generated by your matrices. A subspace I of ℂ^m is simultaneously invariant under every S∈S⊆A exactly when I is an A–submodule of ℂ^m, and then of course any linear subspace S of
 S(I) :...

Finding Common Invariant Subspaces of a Span of Matrices

This is an interesting linear algebra problem about determining pairs of subspaces with specific invariance properties.

Understanding the Problem

You have linearly-independent complex matrices and want to find all pairs where:

• is a linear subspace of $\mathcal{G} = \text{span...