By Shabana A.A.
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That is, the trace is indeed invariant under coordinate transformation. In addition to the trace, the determinant of A is the same as the that is, jAj ¼ A . This important result can also be obtained in determinant of A, the case of second-order tensors using the facts that the determinant of an orthogonal matrix is equal to ±1 and the determinant of the product of matrices is equal to the product of the determinants of these matrices. If A and B are second-order tensors, the double product or double contraction is defined as P3 qj q, j¼1 dqj a A : B ¼ trðAT BÞ ð1:50Þ Using the properties of the trace, one can show that A : B ¼ trðAT BÞ ¼ trðBAT Þ ¼ trðBT AÞ ¼ trðABT Þ ¼ 3 X aij bij ð1:51Þ i, j¼1 where aij and bij are, respectively, the elements of the tensors A and B.
The Householder transformation operates on the columns of the matrix A to produce a set of orthonormal vectors. For example, if A is a 3 Â 3 matrix, one can make the first column a unit vector and use this unit vector with the other two columns to produce an orthogonal triad that consists of three orthonormal vectors. This triad defines a coordinate system and the orthogonal matrix Q. 6 D’Alembert’s Principle 27 to the upper triangular matrix B defined as B ¼ QT A. The way the orthogonal matrix Q is defined here gives a physical interpretation for the QR decomposition of 3 Â 3 matrices.
In a similar manner to the third-order tensor, a Fourth-order tensor F can be defined as F ¼ ðu1 u2 u3 u4 Þ ¼ 3 X i,j,k,l¼1 f ijkl ii ij ik il ð1:82Þ where um , m ¼ 1, 2, 3, 4 is an arbitrary vector. As in the case of third-order tensors, one can write u4m ðu1 u2 u3 Þ ¼ Fim , where u4m is the mth component of the vector u4 . It can then be shown that the coefficients fijkl can be written as ÿ Á f ijkl ¼ ii ij : F : ðik il Þ ð1:83Þ More generally, if v is an arbitrary vector, one has Fv ¼ ðu1 u2 u3 u4 Þv ¼ ðv Á u4 Þðu1 u2 u3 Þ ð1:84Þ which results in a third-order tensor.
Computational Continuum Mechanics by Shabana A.A.