Vectors_Tensors_04_Matrices_and_Index_Notation

Vectors_Tensors_04_Matrices_and_Index_Notation - Section...

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Section 1.4 Solid Mechanics Part III Kelly 22 1.4 Matrices and Element Form 1.4.1 Matrix – Matrix Multiplication In the next section, §1.5, regarding vector transformation equations, it will be necessary to multiply various matrices with each other (of sizes 1 3 × , 3 1 × and 3 3 × ). It will be helpful to write these matrix multiplications in a short-hand element form. First, it has been seen that the dot product of two vectors can be represented by [ ][ ] v u T , or i i v u . Similarly, the matrix multiplication [ ][ ] T v u gives a 3 3 × matrix with element form j i v u or, in full, 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 v u v u v u v u v u v u v u v u v u This type of matrix represents the tensor product of two vectors, written in symbolic notation as v u (or simply uv ). Tensor products will be discussed in detail in a later section. Next, the matrix multiplication [] []
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Vectors_Tensors_04_Matrices_and_Index_Notation - Section...

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