Compute the trace of a square matrix with property verification, eigenvalue connection, diagonal contribution bars, and two-matrix mode for cyclic property testing.
The trace of a square matrix is the sum of its diagonal elements. Despite its simplicity, the trace is one of the most important scalar invariants in linear algebra, appearing throughout mathematics, physics, and machine learning.
The trace satisfies remarkable algebraic properties. It is linear: tr(A + B) = tr(A) + tr(B) and tr(kA) = k·tr(A). It is invariant under transposition: tr(Aᵀ) = tr(A). Most strikingly, it satisfies the cyclic property: tr(AB) = tr(BA), even when AB ≠ BA. This extends to any number of factors: tr(ABC) = tr(BCA) = tr(CAB).
Perhaps the most profound connection is to eigenvalues: the trace always equals the sum of all eigenvalues (counted with multiplicity). Combined with the fact that the determinant equals the product of eigenvalues, the trace and determinant together capture key spectral information.
In physics, the trace appears in the stress-energy tensor, moment of inertia calculations, and quantum mechanics (where the trace of the density matrix equals 1). In machine learning, the Frobenius norm satisfies ‖A‖²_F = tr(AᵀA), connecting the trace to matrix norms used everywhere in optimization and regularization.
While summing diagonal entries seems straightforward, the trace’s importance lies in its algebraic properties — verifying the cyclic identity tr(AB) = tr(BA), confirming eigenvalue sums, and computing the Frobenius norm via tr(AᵀA). This calculator sums the diagonal instantly for matrices up to 5×5, supports optional two-matrix trace identities (tr(A + B), tr(AB), tr(BA)), shows each diagonal element’s contribution, and summarizes the six key trace properties. It turns a simple sum into a full property audit.
tr(A) = Σᵢ aᵢᵢ = a₁₁ + a₂₂ + … + aₙₙ — the sum of diagonal elements.
Result: tr(A) = 4 + 5 + 6 = 15
The trace is the sum of the diagonal elements: 4 + 5 + 6 = 15. This also equals the sum of the eigenvalues of A.
The trace is **linear**: tr(A + B) = tr(A) + tr(B) and tr(cA) = c·tr(A). It is **cyclic**: tr(ABC) = tr(BCA) = tr(CAB) (but ≠ tr(ACB) in general). These two properties make the trace one of the few truly coordinate-invariant quantities of a matrix — similar matrices P⁻¹AP always have the same trace as A. The trace of the identity I_n is n, and the trace of a nilpotent matrix is always 0 because all its eigenvalues are zero.
The trace equals the **sum of the eigenvalues** (counted with algebraic multiplicity): tr(A) = λ₁ + λ₂ + … + λₙ. For a 2×2 matrix, this gives the shortcut: eigenvalues are (λ₁, λ₂) where λ₁ + λ₂ = tr(A) and λ₁λ₂ = det(A). Similarly, the characteristic polynomial p(λ) = λⁿ − tr(A)λⁿ⁻¹ + … + (−1)ⁿ det(A). This makes the trace the simplest spectral invariant to compute.
The **Frobenius norm** is defined via the trace: ‖A‖_F² = tr(AᵀA) = Σaᵢⱼ², which connects matrix norms to eigenvalues. In machine learning, the trace appears in **dimensionality reduction** (maximizing tr(WᵀSW) in PCA), **regularization** (nuclear norm = trace of singular values), and **kernel methods** (tr(K) is the total variance captured). In physics, the trace of a density matrix equals 1 (probability conservation), and in differential geometry, the Ricci scalar is a trace of the curvature tensor.
The trace is the sum of the diagonal elements of a square matrix. It is denoted tr(A) and equals a₁₁ + a₂₂ + … + aₙₙ.
tr(AB) = tr(BA) for any matrices A and B where both products are defined and square. This extends to multiple factors: tr(ABC) = tr(BCA) = tr(CAB).
The trace equals the sum of all eigenvalues (counted with multiplicity). This is because the characteristic polynomial has trace as the coefficient of the second-highest term.
No. The trace is only defined for square matrices because non-square matrices do not have a main diagonal that runs from (1,1) to (n,n).
tr(Iₙ) = n, since the identity has 1s on the diagonal. This confirms that the trace counts the "size" of the identity transformation.
The Frobenius norm satisfies ‖A‖²_F = tr(AᵀA) = sum of all squared elements. This connects the trace to a widely used matrix norm in optimization and machine learning.