Matrix Trace Calculator
Calculate the trace of square matrices by summing diagonal elements
Matrix Input
Trace is only defined for square matrices
Matrix Elements
Diagonal elements are highlighted in purple. Enter decimal numbers (e.g., 0.5, -2.3)
Quick Examples
Matrix Trace Results
Matrix Properties
- Diagonal Matrix:No
- Identity Matrix:No
- Symmetric Matrix:No
Trace Properties
Formula: tr(A) = a₁₁ + a₂₂ + ... + aₙₙ
Linear Property: tr(A + B) = tr(A) + tr(B)
Scalar Property: tr(kA) = k·tr(A)
Step-by-Step Calculation
Trace Analysis
Matrix Visualization
Diagonal elements (trace components) are highlighted in purple
How to Calculate Trace
Identify Diagonal
Find elements where row = column
Sum Elements
Add all diagonal elements
Result
The sum is your matrix trace
Trace Properties
Linear: tr(A + B) = tr(A) + tr(B)
Scalar: tr(kA) = k·tr(A)
Cyclic: tr(AB) = tr(BA)
Transpose: tr(Aᵀ) = tr(A)
Eigenvalues: tr(A) = Σλᵢ
Identity: tr(I) = n
Understanding Matrix Trace
What is Matrix Trace?
The trace of a square matrix is the sum of all its diagonal elements. It's denoted as tr(A) and represents one of the most fundamental invariants of a matrix. The trace provides important information about the matrix's properties and its eigenvalues.
Formula
tr(A) = a₁₁ + a₂₂ + a₃₃ + ... + aₙₙ
Sum of diagonal elements from top-left to bottom-right
Important Properties
- Linearity: The trace is a linear operator
- Cyclicity: tr(ABC) = tr(BCA) = tr(CAB)
- Invariance: tr(A) = tr(Aᵀ) for transpose
- Eigenvalue Sum: tr(A) equals sum of eigenvalues
Applications
- •Eigenvalue calculations and characteristic polynomials
- •Matrix similarity and invariant theory
- •Linear algebra and differential geometry
- •Quantum mechanics and statistical mechanics