Determinant AD Notes

1. Determinant

Forward Definition

For

d = \det(A), \qquad A \in \mathbb{C}^{N \times N},

Jacobi’s formula gives

\dot{d} = \det(A) \cdot \operatorname{tr}(A^{-1}\dot{A}).

Reverse Rule

Given a cotangent \bar{d}:

  • real case:

\bar{A} = \bar{d} \cdot \det(A) \cdot A^{-\mathsf{T}}

  • complex case:

\bar{A} = \overline{\bar{d} \cdot \det(A)} \cdot A^{-\mathsf{H}}.

Singular matrix handling

The inverse formula fails at singular matrices, but the adjugate interpretation still makes sense:

  • rank N-1: the adjugate is rank 1 and can be reconstructed from an SVD
  • rank \le N-2: the adjugate vanishes

PyTorch’s linalg_det_backward handles this regime by reconstructing the leave-one-out singular-value products together with the orientation/phase factor coming from U and V^{\mathsf{H}}.

2. slogdet

Forward Definition

(\operatorname{sign}, \operatorname{logabsdet}) = \operatorname{slogdet}(A).

If w = \operatorname{tr}(A^{-1}\dot{A}), then in the complex case

\dot{\operatorname{logabsdet}} = \operatorname{Re}(w), \qquad \dot{\operatorname{sign}} = i \operatorname{Im}(w)\operatorname{sign}.

Reverse Rule

For the differentiable log-magnitude path:

  • real case:

\bar{A} = \overline{\operatorname{logabsdet}} \cdot A^{-\mathsf{T}}

  • complex case:

\bar{A} = g \cdot A^{-\mathsf{H}}, \qquad g = \overline{\operatorname{logabsdet}} - i \operatorname{Im}(\overline{\operatorname{sign}}^* \operatorname{sign}).

slogdet is not differentiable at singular matrices because \operatorname{logabsdet} = -\infty there.

Implementation Correspondence

  • tenferro-rs/docs/AD/det.md keeps both det and slogdet in one note and discusses the singular adjugate path explicitly.
  • PyTorch’s linalg_det_jvp, linalg_det_backward, slogdet_jvp, and slogdet_backward implement the same split and use solves rather than explicit inverses.

Verification

  • compare primal det(A) and slogdet(A) with direct evaluation
  • compare JVP/VJP against finite differences away from singularity

References

  1. C. G. J. Jacobi, “De formatione et proprietatibus determinantium,” 1841.
  2. M. B. Giles, “An extended collection of matrix derivative results for forward and reverse mode AD,” 2008.

DB Families

### det/identity

The DB publishes the determinant value directly.

### slogdet/identity

The DB publishes the differentiable slogdet observable directly.