Build an orthogonal matrix from a real-parameter vector

Constructs a \(k \times k\) orthogonal matrix \(O\) by exponentiating a skew-symmetric matrix \(S\) built by assigning its lower-triangular entries from the input vector. Specifically, the function sets \(S_{ij}\) (for \(i>j\)) from `entries`, mirrors it to enforce \(S = L - L^\top\), and returns `expm::expm(S)`, which is guaranteed orthogonal because \(\exp(S)\) is orthogonal whenever \(S\) is real skew-symmetric.

Usage

build_orthogonal_matrix(entries)

Arguments

entries

A numeric vector (possibly `NULL`). If `NULL`, returns the `1 x 1` identity. Otherwise, its length must be \(n = k(k-1)/2\) for some integer \(k \ge 2\), supplying the strictly lower-triangular entries of a skew-symmetric generator.

Value

A `k x k` orthogonal matrix. If `entries` is `NULL`, returns `matrix(1, 1, 1)` (identity).

Details

The dimension `k` is inferred from `length(entries)` via the relation \(n = k(k-1)/2\), so `length(entries)` must equal a triangular number.

- Dimension inference uses \(k = \frac{1 + \sqrt{1 + 8n}}{2}\) where \(n = \text{length(entries)}\). If \(k\) is not an integer, the input is invalid and an error is thrown. - Orthogonality follows from the fact that \(S^\top = -S\) implies \(\exp(S)^\top \exp(S) = I\). - Note the function actually returns a special orthogonal matrix, i.e., the determinant is +1.

Examples

# 1x1 identity (NULL input)
build_orthogonal_matrix(NULL)
#>      [,1]
#> [1,]    1

# 2x2 orthogonal matrix from one parameter
O2 <- build_orthogonal_matrix(0.0)
all.equal(t(O2) %*% O2, diag(2)) # should be TRUE
#> [1] TRUE

# 3x3 example: length(entries) = 3 (= 3*2/2), so k = 3
O3 <- build_orthogonal_matrix(c(0.1, -0.2, 0.3))
all.equal(t(O3) %*% O3, diag(3), tolerance = 1e-10)
#> [1] TRUE