2.4 Some Properties of Tensors
31
Table 2.5 Formulas for
determinant (n = dimension
of the space)
det I = 1
det T T = det T
det(φT) = φ n det T
det(T · U) = det T det U
∂ det T/∂T = (det T) T −T
Table 2.6 Formulas for
inverse
(T T ) −1 = (T −1 ) T ≡ T −T
(T · U) −1 = U −1 · T −1
(T −1 ) m = (T m ) −1 ≡ T −m
2.4.2 Identity Tensor and Inverse
In matrix algebra, the identity matrix consists of ones on the diagonal and zeros
elsewhere. Correspondingly, the identity tensor can be written in the form
I = δ ij e i e j = e i e i ,
(2.23)
which satisfies the relations a · I = I · a = a and T · I = I · T = T. This can be
verified, for example, by the manipulations
a · I = (a i e i ) · (e j e j ) = a i (e i · e j )e j
= a i δ ij e j = a i e i = a.
The inverse of the tensor T is defined by the relation
T · T −1 = T −1 · T = I,
(2.24)
where superscript −1 denotes the inverse. The inverse can be computed from the
matrix representation through the equation
T
−1
=
[cof T] T
det T
,
(2.25)
where det T is the determinant of T and cof T is the matrix of cofactors. Some useful
relations for the inverse are given in Table 2.6.
2.4.3 Eigenvalues and Eigenvectors
An eigenvector of the tensor T is a vector that maintains its direction when
transformed by T. Mathematically, this can be stated in the form of the eigenvalue
problem
T · a = λa