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Theory Manual Version 3.6
 Subsection A.1.10: Determinant of a Tensor Up Section A.1: Second-Order Tensors Subsection A.1.12: Transformation Laws for Cartesian Components of Vectors and Tensors 

A.1.11 Orthogonal Tensor

An orthogonal tensor is a linear transformation which preserves the length of a vector and the angle between vectors. Thus, by definition,
for any vectors and . It follows from this definition and the definition of the dot product of vectors ( , that But , which implies that . Since and are arbitrary, an orthogonal tensor must satisfy . In indicial form, , and in matrix form, .
Note that implies that , i.e., the transpose of an orthogonal tensor is equal to its inverse, since . It follows that The determinant of an orthogonal tensor is given by Here, is the orthonormal basis resulting from the transformation of by . If maintains the handedness of (e.g., if both and form a right-handed basis), then and is called a proper orthogonal transformation (also equivalent to a rigid body rotation). Otherwise, in the case of a reflection which reverses the handedness of the basis vectors, and is called improper (e.g., ).
 Subsection A.1.10: Determinant of a Tensor Up Section A.1: Second-Order Tensors Subsection A.1.12: Transformation Laws for Cartesian Components of Vectors and Tensors