Matrix Trace Calculator
Calculate the trace of a square matrix (sum of diagonal elements).
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Matrix Trace
The trace of a square matrix A is the sum of its diagonal elements:
\[ \text{tr}(A) = \sum_{i=1}^{n} a_{ii} \]
Properties:
- \(\text{tr}(A + B) = \text{tr}(A) + \text{tr}(B)\)
- \(\text{tr}(cA) = c \cdot \text{tr}(A)\) for scalar c
- \(\text{tr}(AB) = \text{tr}(BA)\)
- \(\text{tr}(A^T) = \text{tr}(A)\)
- \(\text{tr}(A^{-1}) = \frac{1}{\det(A)} \text{tr}(\text{adj}(A))\)
Applications:
- Characteristic polynomial coefficients
- Eigenvalue sum
- Quantum mechanics (density matrices)
- Statistics (covariance matrices)
- Machine learning (feature selection)
Special Cases:
- Identity matrix: tr(I) = n
- Zero matrix: tr(0) = 0
- Diagonal matrix: tr(D) = sum of diagonal elements
- Projection matrix: tr(P) = rank(P)
Geometric Interpretation:
- Sum of eigenvalues
- Invariant under similarity transformations
- Related to matrix determinant
- Used in matrix decomposition