Determinant

2nd order tensors

Calculation

The determinants for 2nd order tensors a\boldsymbol{ a} in 2d and 3d are

2d: det(a)=a11a22−a21a123d: det(a)=a11[a22a33−a32a23]−a12[a21a33−a31a23]+a13[a21a32−a31a22]\begin{aligned} \text{2d: }\mathrm{det}(\boldsymbol{ a}) &= a_{11}a_{22} - a_{21}a_{12} \\ \text{3d: }\mathrm{det}(\boldsymbol{ a}) &= a_{11}\left[a_{22}a_{33} - a_{32}a_{23}\right] - a_{12}\left[a_{21}a_{33} - a_{31}a_{23}\right] + a_{13}\left[a_{21}a_{32} - a_{31}a_{22}\right] \end{aligned}

as for 2x2 and 3x3 matrices.

Properties

  • det(aT)=det(a)\mathrm{det}(\boldsymbol{ a}^{\mathrm{T}}) = \mathrm{det}(\boldsymbol{ a})

  • det(ab)=det(a)det(b)\mathrm{det}(\boldsymbol{ a}\boldsymbol{ b}) = \mathrm{det}(\boldsymbol{ a})\mathrm{det}(\boldsymbol{ b})

  • det(a−1)=1/det(a)\mathrm{det}(\boldsymbol{ a}^{-1}) = 1/\mathrm{det}(\boldsymbol{ a})

  • det(ka)=kndet(a)\mathrm{det}(k\boldsymbol{ a}) = k^n \mathrm{det}(\boldsymbol{ a}) (where nn is the dimension)

Differentiation

∂∂a[det(a)]=det(a)a−T\begin{aligned} \frac{\partial }{\partial \boldsymbol{ a}}\left[\mathrm{det}(\boldsymbol{ a})\right] = \mathrm{det}(\boldsymbol{ a})\boldsymbol{ a}^{-\mathrm{T}} \end{aligned}

4th order tensors

2d: det(A)=12!εi1j1εi2j2εi3j3εi4j4Ai1i2i3i4Aj1j2j3j33d: det(A)=13!εi1j1k1εi2j2k2εi3j3k3εi4j4k4Ai1i2i3i4Aj1j2j3j3Ak1k2k3k4\begin{aligned} \text{2d: }\mathrm{det}(\boldsymbol{ A}) &= \frac{1}{2!} \varepsilon_{i_1 j_1} \varepsilon_{i_2 j_2} \varepsilon_{i_3 j_3} \varepsilon_{i_4 j_4} A_{i_1 i_2 i_3 i_4} A_{j_1 j_2 j_3 j_3} \\ \text{3d: }\mathrm{det}(\boldsymbol{ A}) &= \frac{1}{3!} \varepsilon_{i_1 j_1 k_1} \varepsilon_{i_2 j_2 k_2} \varepsilon_{i_3 j_3 k_3} \varepsilon_{i_4 j_4 k_4} A_{i_1 i_2 i_3 i_4} A_{j_1 j_2 j_3 j_3} A_{k_1 k_2 k_3 k_4} \end{aligned}
© Knut Andreas Meyer.
Last modified: April 22, 2024.
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