Gaussian and ordered inverse operations

Gaussian and ordered inverse operations

The supplied Gaussian constructions retain their actual finite coordinate spaces, precision matrices, means and support windows. In particular a nonsymmetric auxiliary matrix is not identified with its symmetric covariance. If the source uses

Searchable transcription

C = (P + P ∗)/2,
μ = −P ∗b,

then both and remain in the realized history. For any comparison vector and the exact identity is

when. Indeed, so the right side is. Replacing by generally changes this identity. A coordinate isomorphism transports the mean as a vector and covariance as a form with the stated metric; the original Gaussian integral and all normalized insertions then recover by change of variables, with its Jacobian retained. No probability law is assigned to a nonsymmetric covariance.

For the source’s approximate inverse of, put. On its original convergence domain, for example 1 in the stated submultiplicative norm,

Searchable transcription

C−1 −A =
X
k≥1
(−S)kA.

To prove this, gives . The finite geometric identity has residual , whose norm tends to zero. Subtracting the zeroth term gives the displayed ordered series. The terminal factor cannot be removed or commuted through without an additional identity. Under an isomorphic coordinate readout each finite ordered product conjugates exactly, and the norm-convergent series follows in the transported norm.

Local determinant diagnostics retain the original supported contribution together with every complementary arm required by the complete determinant identity. A diagnostic contribution alone is not the full log determinant. Likewise a quiet-coordinate condition is evaluated in the actual source coordinates: under it becomes the condition on, not the same componentwise bound on. Orthogonal invariance of a Euclidean norm does not imply invariance of a coordinatewise box. All whole-color-block restrictions and source window conditions are therefore retained before inverse or determinant transport.