Matrix Trace Calculator – Find tr(A) Online

Calculate the trace of any square matrix with our free online matrix trace calculator. Find the sum of diagonal elements instantly with step-by-step display.

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Enter matrix values (each row on a new line, values separated by spaces or commas)

Current matrix: 3x3 | Detected from your input

Understanding Matrix Trace

The trace of a square matrix is simply the sum of its diagonal elements – those running from the top-left corner to the bottom-right. Despite its simplicity, the trace appears throughout mathematics and physics, from quantum mechanics to differential equations.

The trace has remarkable properties: it's invariant under similarity transformations, equals the sum of eigenvalues, and satisfies tr(AB) = tr(BA) even when AB ≠ BA. These properties make it a powerful tool in theoretical work.

Trace Formula and Properties

Definition

For A = [aᵢⱼ], tr(A) = a₁₁ + a₂₂ + ... + aₙₙ = Σ aᵢᵢ

Linearity

tr(A + B) = tr(A) + tr(B)

tr(cA) = c·tr(A)

Cyclic Property

tr(AB) = tr(BA)

tr(ABC) = tr(BCA) = tr(CAB)

Transpose

tr(Aᵀ) = tr(A)

Same diagonal elements

Worked Examples

Example 1: 2x2 Matrix

A = [[1, 2], [3, 4]]

Diagonal elements: 1, 4

tr(A) = 1 + 4 = 5

Example 2: 3x3 Matrix

A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

Diagonal elements: 1, 5, 9

tr(A) = 1 + 5 + 9 = 15

Example 3: Identity Matrix

I₃ = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

Diagonal elements: 1, 1, 1

tr(I₃) = 1 + 1 + 1 = 3

For any n×n identity: tr(Iₙ) = n

Example 4: Diagonal Matrix

D = [[5, 0, 0], [0, 3, 0], [0, 0, 2]]

For diagonal matrices, trace = sum of diagonal entries

tr(D) = 5 + 3 + 2 = 10

Quick Fact

The trace equals the sum of eigenvalues! This deep connection links a simple arithmetic operation (adding diagonal elements) to the fundamental spectral properties of a matrix. In quantum mechanics, the trace of a density matrix always equals 1, representing total probability.

Frequently Asked Questions

Why is it called "trace"?

The term comes from the German "Spur" (track or trace), used by mathematicians in the early 20th century. It suggests the diagonal leaves a "trace" through the matrix from corner to corner.

Can non-square matrices have a trace?

No. The trace requires a main diagonal from top-left to bottom-right, which only exists for square matrices. Rectangular matrices don't have this complete diagonal.

What's the trace of a zero matrix?

Zero! All diagonal elements are 0, so their sum is 0. This holds for any size zero matrix.

How is trace used in physics?

In quantum mechanics, the trace gives expectation values and probabilities. The trace of the density matrix equals 1 (total probability). In statistical mechanics, partition functions involve traces of operators.

Is trace the same as determinant?

No, they're different. Trace is the sum of diagonal elements (and eigenvalues). Determinant is the product of eigenvalues. Both are important matrix invariants but capture different properties.

What's the trace of AB vs BA?

They're equal! tr(AB) = tr(BA) even though AB ≠ BA in general. This cyclic property extends: tr(ABC) = tr(BCA) = tr(CAB), but not necessarily tr(ACB).

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