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Matrix Trace Calculator

Compute trace (sum of diagonal elements) for 2x2 and 3x3 matrices

Enter 2x2 Matrix

Trace Formula

tr(A) = a11 + a22 + ... + ann

tr(A) = sum of eigenvalues

tr(AB) = tr(BA)

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

The trace of a square matrix is the sum of its diagonal elements. It equals the sum of eigenvalues and has properties like tr(AB)=tr(BA), making it useful in quantum mechanics, statistics, and numerical analysis.

⚠ Trace is only defined for square matrices (n×n). Rectangular matrices don't have a trace.

What is Matrix Trace?

The trace is a simple but important matrix operation. Just sum the diagonal entries. Despite its simplicity, it has deep connections to eigenvalues and many useful properties. tr(AB)=tr(BA) is especially important in physics and engineering.

Diagonal Elements

Elements where row index = column index: a11, a22, a33...

Sum of Eigenvalues

tr(A) = λ1 + λ2 + ... + λn. Characteristic polynomial coefficient.

tr(AB)=tr(BA)

Cyclic property. Order matters for product but not for trace.

Linear Property

tr(kA + mB) = k tr(A) + m tr(B). Trace is linear.

Teaching Example: A = [[1,2],[3,4]]
1. Diagonal elements: 1, 4
2. tr(A) = 1 + 4 = 5
3. Check eigenvalues: λ1=5.372, λ2=-0.372, sum=5 ✓

Applications

Quantum Mechanics Statistics Numerical Analysis Graph Theory Physics

Frequently Asked Questions

What is matrix trace?▼

Sum of diagonal elements: tr(A) = a11 + a22 + ... + ann

Trace properties?▼

tr(AB)=tr(BA), tr(A+B)=tr(A)+tr(B), tr(kA)=k tr(A)

Trace equals eigenvalues sum?▼

Yes! tr(A) = λ1 + λ2 + ... + λn (sum of eigenvalues)

Trace for square matrix only?▼

Yes, trace is only defined for square matrices (n×n).

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