What is the trace of a matrix?

The trace is the sum of the elements on the main diagonal of a square matrix — that is, the elements where the row index equals the column index. For a 3×3 matrix with diagonal entries a, b, c, the trace is a + b + c.

Why must the matrix be square?

The trace is only defined for square matrices because the main diagonal connects corresponding row and column positions. Non-square matrices do not have a well-defined main diagonal spanning both dimensions.

Is the trace related to eigenvalues?

Yes. The trace of a matrix equals the sum of its eigenvalues (counted with multiplicity). This is a key identity used in spectral analysis and quantum mechanics.

What is the trace of the identity matrix?

The trace of an n×n identity matrix is n, because every diagonal entry is 1 and all off-diagonal entries are 0.

Can the trace be negative?

Yes. If the diagonal elements include negative numbers that outweigh any positive ones, the trace will be negative.

Does the trace change if I reorder the rows and columns?

Similar matrices (related by a change of basis) have the same trace, but arbitrarily reordering rows without a matching column reorder will change it. The trace is invariant under similarity transformations, not arbitrary permutations.

Matrix Trace Calculator

Name: Matrix Trace Calculator
Availability: InStock
Author: Nham Vu

Enter values into a square matrix grid (up to 5×5) and instantly compute its trace — the sum of the main diagonal elements.

Matrix Input

Matrix size (n × n)

Result

Trace (tr A)

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Step-by-Step

Enter matrix values to see the working.

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Summary

Enter values into a square matrix grid (up to 5×5) and instantly compute its trace — the sum of the main diagonal elements.

How it works

Select the matrix size (2×2 up to 5×5) using the size buttons.
Enter numeric values into each cell of the grid.
The trace is computed automatically as you type.
The step-by-step panel shows which diagonal cells were summed.
Click Reset to clear the grid and start over.

Use cases

Verify hand-computed matrix traces in homework or exams.
Quickly check the sum of eigenvalues via the trace identity.
Validate covariance matrix diagonals in statistics.
Debug linear algebra code by cross-checking intermediate results.
Teach or learn the trace concept with instant visual feedback.
Use as a sanity check when computing matrix similarity transforms.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu