Introduction to hlrhotrix

What is an hl-rhotrix?

An hl-rhotrix (heartless rhotrix, or even-dimensional rhotrix) is a mathematical structure introduced by Isere (2018) as a rhotrix from which the central heart entry has been removed. Unlike classical (odd-dimensional) rhotrices, hl-rhotrices have even cardinality and a rich symmetric structure.

The basal hl-rhotrix of dimension 2 has the rhomboidal form:

    a11
  a21  a12
    a22

where a11, a22 lie on the vertical axis (diagonal) and a21, a12 are symmetric entries.

All operations follow the Robust Multiplication Method (RMM) of Isere & Utoyo (2025).

Creating hl-rhotrices

Dimension 2

R <- make_R2(a11 = 5, a21 = 3, a12 = 6, a22 = 2)
print(R)
#> 
#> === hl-Rhotrix (dim = 2) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [5, 2],  sym = [3, 6]

Dimension 4

The 4-dimensional hl-rhotrix decomposes into two R2 principal minors (A21, A22) and one M2 minor matrix (M21) — see Theorem 3.1.

# Example 3.2 of the manuscript
A <- make_R4(
  a11 = -2,
  a21 =  3, c11 = 4,  a12 = 1,
  a31 =  1, c21 = 2,  c12 = -1, a13 = 0,
  a32 =  1, c22 = 1,  a23 = 0,
  a33 =  3
)
print(A)
#> 
#> === hl-Rhotrix (dim = 4) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [-2, 3],  sym = [1, 0]
#>   A2: diag = [4, 1],  sym = [2, -1]
#> 
#> Minor Matrices (M2):
#>   M1: [[3, 1], [1, 0]]

Dimension 6

R6 <- make_R6(
  a11=1, a21=2, c11=3, a12=4,
  a31=5, c21=6, a22=7, c12=8, a13=9,
  a41=2, c31=3, a32=4, a23=5, c13=6, a14=7,
  a42=1, c32=2, a33=3, c23=4, a24=5,
  a43=6, c33=7, a34=8, a44=9
)
print(R6)
#> 
#> === hl-Rhotrix (dim = 6) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [1, 9],  sym = [2, 7]
#>   A2: diag = [3, 7],  sym = [3, 6]
#>   A3: diag = [7, 3],  sym = [4, 5]
#> 
#> Minor Matrices (M2):
#>   M1: [[5, 9], [1, 5]]
#>   M2: [[2, 4], [6, 8]]
#>   M3: [[6, 8], [2, 4]]

Determinant

The determinant decomposes as (Definition 3.3):

\[\det(R_n) = \det\!\Bigl(\prod_k A_{2k}\Bigr) \otimes \det\!\Bigl(\prod_l M_{2l}\Bigr)\]

The sign of the M2 block is \(-1\) when there is exactly one M2 minor, \(+1\) otherwise (Definition 3.4).

det_hl(R)    # -8   (Example 3.3)
#> [1] -8
det_hl(A)    # -36  (Example 3.2)
#> [1] -36
det_hl(R6)   # 15360
#> [1] 15360

Theorem 3.4: swapping all symmetric entries negates the determinant.

B <- make_R4(
  a11=-2, a21=1, c11=4, a12=3,
  a31=0, c21=-1, c12=2, a13=1,
  a32=0, c22=1, a23=1, a33=3
)
det_hl(B) == -det_hl(A)
#> [1] TRUE

Adjoint

adj_hl(R)
#> 
#> === hl-Rhotrix (dim = 2) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [2, 5],  sym = [-3, -6]
adj_hl(A)
#> 
#> === hl-Rhotrix (dim = 4) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [18, -12],  sym = [-6, 0]
#>   A2: diag = [-6, -24],  sym = [12, -6]
#> 
#> Minor Matrices (M2):
#>   M1: [[0, 36], [36, 108]]

Inverse

inv_hl(R)
#> 
#> === hl-Rhotrix (dim = 2) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [-0.25, -0.625],  sym = [0.375, 0.75]
inv_hl(A)
#> 
#> === hl-Rhotrix (dim = 4) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [-0.5, 0.3333],  sym = [0.1667, 0]
#>   A2: diag = [0.1667, 0.6667],  sym = [-0.3333, 0.1667]
#> 
#> Minor Matrices (M2):
#>   M1: [[0, -1], [-1, -3]]

Verify \(A \cdot A^{-1} = I\) on each R2 block:

Ai <- inv_hl(A)
for (k in seq_along(A$A)) {
  Ak <- matrix(c(A$A[[k]]["diag","col1"], A$A[[k]]["sym","col1"],
                 A$A[[k]]["sym","col2"],  A$A[[k]]["diag","col2"]), nrow=2)
  Ik <- matrix(c(Ai$A[[k]]["diag","col1"], Ai$A[[k]]["sym","col1"],
                 Ai$A[[k]]["sym","col2"],  Ai$A[[k]]["diag","col2"]), nrow=2)
  cat(sprintf("Block A%d * inv(A)%d:\n", k, k))
  print(round(Ak %*% Ik, 8))
}
#> Block A1 * inv(A)1:
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> Block A2 * inv(A)2:
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1

Eigenvalues

For a 2-dimensional hl-rhotrix, the characteristic polynomial is \(\Delta(t) = t^2 + \mathrm{tr}(A)\,t + \det(A)\) (Definition 3.8).

# Example 3.3
trace_hl(R)          # 7
#> [1] 7
char_poly_hl(R)      # t^2 + 7t - 8
#> $coefficients
#>  constant    linear quadratic 
#>        -8         7         1 
#> 
#> $poly_string
#> [1] "t^2 + (7)*t + (-8)"
#> 
#> $trace
#> [1] 7
#> 
#> $determinant
#> [1] -8
eigenvalues_hl(R)    # -8 and 1
#> [1]  1 -8

For higher dimensions, eigenvalues are computed per principal R2 minor:

eigenvalues_hl_high(A)
#> [[1]]
#> [[1]]$minor
#> [1] "A21"
#> 
#> [[1]]$eigenvalues
#> [1]  2 -3
#> 
#> 
#> [[2]]
#> [[2]]$minor
#> [1] "A22"
#> 
#> [[2]]$eigenvalues
#> [1] -2 -3
eigenvalues_hl_high(R6)
#> [[1]]
#> [[1]]$minor
#> [1] "A21"
#> 
#> [[1]]$eigenvalues
#> [1]   0.4772256 -10.4772256
#> 
#> 
#> [[2]]
#> [[2]]$minor
#> [1] "A22"
#> 
#> [[2]]$eigenvalues
#> [1] -0.3095842 -9.6904158
#> 
#> 
#> [[3]]
#> [[3]]$minor
#> [1] "A23"
#> 
#> [[3]]$eigenvalues
#> [1] -0.1010205 -9.8989795

Full summary

summary_hl(R)
#> ============================================================== 
#>   hl-Rhotrix Summary  (dim = 2)
#> ============================================================== 
#> 
#> === hl-Rhotrix (dim = 2) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [5, 2],  sym = [3, 6]
#> 
#> Determinant:  -8
#> 
#> Adjoint:
#> 
#> === hl-Rhotrix (dim = 2) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [2, 5],  sym = [-3, -6]
#> 
#> Inverse:
#> 
#> === hl-Rhotrix (dim = 2) ===
#> 
#> Principal Minor Rhotrices (R2):
#>   A1: diag = [-0.25, -0.625],  sym = [0.375, 0.75]
#> 
#> Trace:               7
#> Characteristic poly: t^2 + (7)*t + (-8)
#> Eigenvalues:         lambda1 = 1,  lambda2 = -8
#> 
#> ==============================================================

Rhomboidal layout (ggplot2)

plot_rhombus_gg() returns a ggplot object showing the rhomboidal structure. Blue nodes/regions are principal R2 minors; amber nodes/regions are M2 minor matrices.

plot_rhombus_gg(2, title = "hl-Rhotrix  (dim = 2)")
#> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
#> ℹ Please use `linewidth` instead.
#> ℹ The deprecated feature was likely used in the hlrhotrix package.
#>   Please report the issue to the authors.
#> This warning is displayed once per session.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.

plot_rhombus_gg(4,
  title    = "hl-Rhotrix  (dim = 4)",
  subtitle = "Minors A21, A22 (R2) and M21 (M2)")

plot_rhombus_gg(6,
  title    = "hl-Rhotrix  (dim = 6)",
  subtitle = "Minors A21, A22, A23 (R2) and M21, M22, M23 (M2)")

The returned object integrates naturally with the ggplot2 ecosystem:

# Export for the manuscript (PDF, 300 dpi)
ggsave("fig_rhombus_dim4.pdf", plot_rhombus_gg(4),
       width = 10, height = 10, units = "cm", dpi = 300)

# Three-panel figure with patchwork
library(patchwork)
plot_rhombus_gg(2) + plot_rhombus_gg(4) + plot_rhombus_gg(6)

# Further customisation
plot_rhombus_gg(4) +
  theme(plot.background = element_rect(fill = "grey98", colour = NA))

References