Compute essential properties of any second-order tensor: trace, determinant, principal invariants, symmetric and skew-symmetric decomposition, Frobenius norm, and characteristic polynomial coefficients. Interactive 3×3 component grid for continuum mechanics, relativity, and engineering analysis.
A second-order tensor in three dimensions is a linear map that transforms vectors into vectors, often represented by a 3×3 matrix. In physics and engineering, tensors describe stress, strain, inertia, diffusion, and curvature. The trace, determinant, and principal invariants (I₁, I₂, I₃) are intrinsic properties independent of coordinate system – making them essential for material modeling and relativity.
For any tensor T, the characteristic equation: det(T - λI) = -λ³ + I₁ λ² - I₂ λ + I₃ = 0, where I₁ = tr(T), I₂ = ½[(tr(T))² - tr(T²)], I₃ = det(T).
Every tensor uniquely splits into symmetric (T(ij)) and skew-symmetric (T[ij]) parts. In continuum mechanics, the symmetric part represents pure deformation (strain), while the skew part corresponds to infinitesimal rotation. The decomposition helps separate volumetric and deviatoric responses, crucial for plasticity and fluid dynamics.
Our calculator computes the trace as sum of diagonal entries. The determinant uses the general 3×3 formula (Sarrus rule). The second invariant is computed via I₂ = ½[(tr T)² - tr(T²)] for numerical stability. Symmetric part: (T + Tᵀ)/2, skew part: (T - Tᵀ)/2. Frobenius norm = sqrt(Σ Tij²). The deviatoric norm uses the deviatoric tensor s = T - (tr(T)/3)I, important for shear-dominant processes.
Consider a stress tensor at a point: σ = [[80, 20, 0], [20, 40, 0], [0, 0, 30]] MPa. Using our tensor calculator, trace = 150 MPa (related to volumetric stress), determinant = 80*40*30 + ... = 76,000 MPa³, and the von Mises equivalent stress is derived from the deviatoric part. The symmetric part is the tensor itself (since symmetric). Engineers rely on invariants to predict yielding independent of coordinate orientation. Such analysis is standard in finite element post-processing.
| Tensor Type / Example | Trace | Determinant | I₂ | Physical Context |
|---|---|---|---|---|
| Identity I₃×₃ | 3 | 1 | 3 | Isotropic reference |
| Diag(2,3,5) | 10 | 30 | 31 | Orthotropic material stiffness |
| Symmetric stress [[50,10,0],[10,30,0],[0,0,20]] | 100 | 50*30*20 - ... ≈ 26000 | 2500 | Plane stress state |
| Skew-symmetric [[0,ω,-ω],[-ω,0,0],[ω,0,0]] | 0 | 0 | 2ω² | Infinitesimal rotation tensor |
Our calculator provides the coefficients of the characteristic polynomial λ³ – I₁ λ² + I₂ λ – I₃ = 0. The roots (eigenvalues) determine principal tensor values. While we do not compute explicit roots for all cases (due to potential complexity), the invariants fully characterize the eigenvalue spectrum up to permutation. For symmetric positive-definite tensors, eigenvalues are real and positive – critical for strain energy functions.