Chris Liu
Math gradudate student at Colorado State University
Simultaneous Sylvester Systems (SSS) and some applications
Chris Liu (chris.liu@colostate.edu)
Colorado State University
June 2025
Joint with James B. Wilson and Joshua Maglione
I bring this waterbottle with me everywhere and will probably lose it once this week!
Center
\( \{ X \in A \;|\; (\forall i) \;XA_i = A_i X \} \)
Centralizer & Module End
\( \{ X \in \mathbb{M}_n(K) \;|\; (\forall i) \; XA_i = A_i X \} \)
Centroid algebra
\( \left\{ (X,Y,Z) \in \mathbb{M}_n(K) \times \mathbb{M}_m(K) \times \mathbb{M}_c(K) \;|\; (\forall i) \; XT_{i} = T_{i}Y = \left(\sum_j Z_{ij}T_j \right)_i \right\} \)
Module hom
\( \{ X \in \mathbb{M}_{n\times m}(K) \;|\; (\forall i) \;XA_i = B_i X \} \)
Adjoint algebra
\( \{ (X,Y) \in \mathbb{M}_n(K) \times \mathbb{M}_m(K) \;|\; (\forall i) \; XT_{i} = T_{i} Y \} \)
Given \(A = \langle A_1,\ldots, A_c \rangle \) a matrix algebra
Given \( T = [T_{1}, \ldots, T_{c}] \) with \(T_{i} \in \mathbb{M}_{n \times m}(K) \) - a tensor
For lists of \(n \times n\) matricies, decades of rich theory to avoid solving linear system of \( n^2 \) variables
Center
Using randomization - Friedl-Ronyai '85, Ivanyos–Rónyai '94, Eberly-Giesbrecht '96,...
Centralizer & Module End
Case of module hom
Module hom
Meataxe - Horton-Parker '84 Holt-Rees '94, Ivanyos-Lux '00, ...
Condensation - Thackray '81, Ryba '90, Lux-Wiegelmann '95, Lübeck-Neunhöffer '01, ...
Peakwords - Lux-Szöke '03
Adjoint algebra
Kronecker modules for \( c = 2 \) - Brooksbank-Maglione-Wilson '17
Centroid algebra
Optimizations for c = 2 case in Magma
A bottleneck example
A_i = \begin{bmatrix}
I_n & C_i & 0 \\
0 & I_n & B_i \\
0 & 0 & I_n
\end{bmatrix}, A = \langle A_1,\ldots,A_m \rangle
Question: Does there exist an element in \( \operatorname{End}_A(M) \) of the form
\( \begin{bmatrix} I_n & \ast & 0 \\ 0 & I_n & \ast \\ 0 & 0 & I_n \end{bmatrix} \)
- Local algebra, no idempotents to split
- Many non-redundant generators as \(A_iA_j\) only adds on the (1,2) and (2,3) blocks.
\( \begin{bmatrix} X & U & W \\ 0 & Y & V \\ 0 & 0 & Z \end{bmatrix} \in \operatorname{End}_A(M)\) satisfies
\begin{bmatrix} X & U & W \\ 0 & Y & V \\ 0 & 0 & Z \end{bmatrix}
\begin{bmatrix}
I_n & A_i & 0\\
0 & I_n & B_i \\
0 & 0 & I_n
\end{bmatrix} =
\begin{bmatrix}
I_n & A_i & 0\\
0 & I_n & B_i \\
0 & 0 & I_n
\end{bmatrix}
\begin{bmatrix} X & U & W \\ 0 & Y & V \\ 0 & 0 & Z \end{bmatrix}
XA_i = A_iY\\
YB_i = B_i Z\\
UB_i = A_iV
Centralizer-like relations need to be satisfied for all \(i \)
Currently, no better ideas than solving \(2n^2\) variable linear systems in \(n^3\) equations
- Not a center, centralizer, or module endomorphism question
- End of the line for algebra splitting and radical layers splitting of original algebra
Our case: \( X = Y = Z = I_n \) and we just need \( U B_i = A_i V\)
Magma example code here showcase slow
Mention difference compared to unitraingular (non-blocked) case or other theory cases where work bottoms out to tiny systems
Create \( B_i \) from \(A_i \)
Create \(B_i\) randomly
(SSS) Simultaneous Sylvester Systems
A_i \in \mathbb{M}_{r \times b}(K)\\
B_i \in \mathbb{M}_{a \times s}(K)\\
C_i \in \mathbb{M}_{a \times b}(K)
X \in \mathbb{M}_{a \times r}(K)\\
Y \in \mathbb{M}_{s \times b}(K)
Given
Find
Such that
(\forall i) \; XA_i + B_iY = C_i
\( i \in \{1,\ldots, c\} \)
Center, module end/hom, adjoint, centroid all instances
Without assuming other structure, there is interwoven striding in augmented matrix of flattened linear system
\begin{bmatrix} A_1 \\ \vdots \\ A_b \end{bmatrix} \otimes I_a
\begin{bmatrix}
B_1 \otimes I_b\\
\vdots\\
B_a \otimes I_b
\end{bmatrix}
The fine print
\( K = \text{GF}(5)\) here and throughout
Bottleneck module end problem is for the below module in Magma
A_i := \begin{bmatrix}
I_n & R_i & T_i \\
0 & I_n & S_i \\
0 & 0 & I_n
\end{bmatrix}, \; M := \text{RModule}([A_1,\ldots, A_{10}])
Yet this system scales like solving system of \(n^2\) variables
How much to solve depends on an invertibility condition
Theorem (L.-Maglione-Wilson)
Given \( A \in K^{r \times b \times c}\), \(B \in K^{a \times s \times c} \), and \(C \in K^{a \times b \times c} \) an instance of SSS
If \( \text{HorizontalJoin}(A(:, 1:b', :) ) \) and \( \text{HorizontalJoin}(B(1:a', :, :) \) are full rank
Then a solution to SSS is found by solving a linear system of \(a'r + b's\) variables followed by backsubstitution
Idea: Avoid augmented matrix altogether with a solve-backsubstitute-check approach
For the roughly cubic ("thick") case, we expect \( a' \) and \( b' \) to be constants as function of \( n \). Reduces solving an \( O(n^2) \) variable system to an \(O(n) \) variable system
\}
Being able to bound \( n' \) to be much smaller than \( n\) is a complexity win
(More precisely)
Solving a linear system of \( 2nn'\) variables
Two one-sided inverses of \(n \times nn'\) matricies
Two products of \( n^2 \times n \) matrix with \( n \times n' \) matrix
Two products of \( n \times nn' \) matrix with \( nn' \times n \) matrix
Instead of solving a system of \(ar + bs \) variables in \(abc \) equations, below are precisely the computations necessary in QuickSylver (bullet point are the variables in code)
- Solving a linear system of \( a'r + b's \) variables
- Optimization: don't take all constraints, check correctness at the end
- One-sided inverses of \( r \times b'c \) and \( s \times a'c \) matrices
- `r_B_prime` and `s_A_prime`
- The product of a \( c(a-a') \times s \) matrix with a \( s \times b' \) matrix
- `sy_U_B_prime_stacked`
- The product of a \( a' \times r \) matrix with a \( r \times c(b-b') \) matrix
- `rx_A_prime_V_stacked`
- The subtraction of two \( (a-a') \times b'c \) matricies
- (I) t_U_B_prime_stacked - sy_U_B_prime_stacked
- The subtraction of two \( (b-b') \times a'c \) matricies
- (II) t_A_prime_V_stacked - rx_A_prime_V_stacked
- The product of a \( (a-a') \times b'c\) matrix with a \( b'c \times r\) matrix
- Multiplying (I) with `r_B_prime_inv`
- The product of a \( (b-b') \times a'c \) matrix with a \( a'c \times s \) matrix
- Multiplying (II) with `s_A_prime_inv`
Magma implementation results
The fine print
- Only provides a specific solution, not full basis of nullspace (tunable!)
- Randomized check of solution correctness (tunable!)
- Bottleneck module end problem is for the below module in Magma
A_i := \begin{bmatrix}
I_n & R_i & T_i \\
0 & I_n & S_i \\
0 & 0 & I_n
\end{bmatrix}, \; M := \text{RModule}([A_1,\ldots, A_{10}])
Clustering and data science application
\( T = [T_1, \ldots, T_c] \) with \(T_i \in \mathbb{M}_{n \times m}(K) \) given linear data from an external source
Solving for the adjoint algebra by a Simultaneous Sylvester Equation and decomposing it clusters original data
Example: The \( (j,k) \) entry of \( T_i \) stores number of occurences user \(i\) at time \(j\) chatted word \(k\) - called the "chatroom tensor" in the literature
- Can hybrid approaches win even more?
- Are there bottleneck cases to try in your problems?
- Solving generalizations of SSS?
Questions?
Find us (Chris Liu, James B. Wilson, and Josh Maglione) at software demo session if interested!
Simultaneous Sylvester System and an application to module endomorphisms
By Chris Liu
