reflection of the matrix elements in the main diagonal of the matrix C Matrix

Reflection of the matrix elements in the main

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reflection of the matrix elements in the main diagonal of the matrix. C. Matrix Multiplication The multiplication of two matrices, A of m × k dimensions and B of k × n dimensions, is defined as an operation that produces a matrix C of m × n dimensions whose C ij entry is the dot product of the i th row of the first matrix A and the j th column of the second matrix B . Hence, if A is a 3 × 2 matrix and B is a 2 × 2 matrix, then their product AB is a 3 × 2 matrix which is given by: AB = A 11 A 12 A 21 A 22 A 31 A 32 B 11 B 12 B 21 B 22 = A 11 B 11 + A 12 B 21 A 11 B 12 + A 12 B 22 A 21 B 11 + A 22 B 21 A 21 B 12 + A 22 B 22 A 31 B 11 + A 32 B 21 A 31 B 12 + A 32 B 22 (45) [17] The indexing of the entries of A T is not standard; the purpose of this is to demonstrate the exchange of rows and columns.
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1.3.3 Matrix Algebra 37 From the above, it can be seen that matrix multiplication is defined only when the number of columns of the first matrix is equal to the number of rows of the second matrix. Matrix multiplication is associative and distributive over a sum of compatible matrices, but it is not commutative in general even if both forms of the product are defined, that is: ( AB ) C = A ( BC ) (46) A ( B + C ) = AB + AC (47) AB 6 = BA (48) As seen above, no symbol is used to indicate the operation of matrix multiplication according to the notation of matrix algebra , i.e. the two matrices are put side by side with no symbol in between. However, in tensor symbolic notation such an operation is usually represented by a dot between the symbols of the two matrices, as will be discussed later in the book. [18] D. Trace of Matrix The trace of a matrix is the sum of its diagonal elements , therefore if a matrix A is given by: A = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 (49) then its trace is given by: tr ( A ) = A 11 + A 22 + A 33 (50) [18] In brief, AB represents an inner product of A and B according to the matrix notation, and an outer product of A and B according to the symbolic notation of tensors, while A · B represents an inner product of A and B according to the symbolic notation of tensors. Hence, AB in the matrix notation is equivalent to A · B in the symbolic notation of tensors.
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1.3.3 Matrix Algebra 38 From its definition, it is obvious that the trace of a matrix is a scalar and it is defined only for square matrices . E. Determinant of Matrix The determinant is a scalar quantity associated with a square matrix . There are several definitions for the determinant of a matrix; the most direct one is that the determinant of a 2 × 2 matrix is the product of the elements of its main diagonal minus the product of the elements of its trailing diagonal, that is: A = A 11 A 12 A 21 A 22 det ( A ) = A 11 A 12 A 21 A 22 = A 11 A 22 - A 12 A 21 (51) The determinant of an n × n ( n > 2 ) matrix is then defined, recursively, as the sum of the products of each entry of any one of its rows or columns times the cofactor of that entry where the cofactor of an entry is defined as the determinant obtained from eliminating the row and column of that entry from the parent matrix with a sign given by ( - 1) i + j with i and j being the indices of the row and column of that entry.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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