Trace

Shipping

matrix_trace

Sum of the diagonal, tr(A) = Σ Aᵢᵢ

Signature

Inputs

aSignal|Arrayrequired— A 2-D matrix.

Outputs

traceScalar— The trace $\operatorname{tr}(A) = \sum_i A_{ii}$.

Description

Trace returns the sum of the main-diagonal entries of a matrix, $tr (A) = \sum_{i} A_{ii}$ , via the MKL LAPACK-backed toolkit. The trace equals the sum of the eigenvalues and is invariant under similarity transforms and cyclic permutations, which makes it a lightweight invariant for checking decompositions and computing quantities like the sum of variances of a covariance matrix.

The input is read as a 2-D matrix; the result is a dimensionless Scalar.

Mathematics

tr (A) = i \sum A_{ii} = i \sum λ_{i}

tr (A B) = tr (B A)

Examples

Trace equals eigenvalue sum

For $A = 411130102$ , the trace output is $4 + 3 + 2 = 9$ — which also equals the sum of $A$ 's eigenvalues, a handy cross-check against matrix_eigh.

Applications

Summing variances (total variance) from a covariance matrix's diagonal.
Cheap invariant to validate eigen-decompositions (trace = Σλ).
Computing quadratic-form and quantum-mechanical expectation traces.
Sanity-checking similarity transforms, which preserve the trace.

Neat

The trace is a similarity invariant, so it survives change-of-basis untouched — a $O(n)$ check on an $O(n^3)$ decomposition.

It equals the eigenvalue sum, giving a free consistency test against matrix_eig / matrix_eigh.

Known issues

Defined only for the main diagonal of a 2-D matrix — for a general axis sum use tensor_reduce or einsum.

A non-square matrix's diagonal sum is well-defined but rarely meaningful; trace is intended for square inputs.