Matrix Multiplication In Index Notation

13 Sep, 2021

B 3 ca 3. I or simply.

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Lambdaalpha _muLambda_alphasigmadelta_mu sigma tag 1 We would like to identify this as matrix multiplication and thus identify the rightmost Lambdain this equation as the inverse.

Matrix multiplication in index notation. The rule for matrix multiplication is. Make the inner index k in this case the same and sum over it Example. We show how to use index notation and sum over row and column indices to perform matrix multiplication.

One free index as here indicates three separate equations. 3 It is simple to see that the index i is incremented between 1 and 3 in equation 3 and the multiplication is carried out. Ca b b 1 b 2 b 3 ca 1 ca 2 ca 3 2 ca i b i.

Multiplication rules are in fact best explained through tensor notation. The index notation for these equations is. If a vector X either a row or column vector has N components then X X i stands for X 1 X 2 X 3 X N.

Multiplication of a matrix by a number is called scalar multiplication. B1 b2 b3 9. IjtobeanRPmatrixThematrixproductAB isde ned onlywhenRNandistheMPmatrixCc ijgivenby c ij XN k1 a ikb kj a i1b1j a i2b2j a iNb Nk Usingthesummationconventionthiscanbewrittensimply c ij a ikb kj wherethesummationisunderstoodtobeovertherepeatedindexkInthecaseofa33 matrixmultiplyinga3 1columnvectorwehave 2 6 4 a11 a12 a13 a21 a22 a23 a31 a32 a33 3 7 5 8.

I i j ij b a x ρ σ 7111 Note the dummy index. Alternative notations X X i X i X i XN i1 X i are used when we want to be explicit about the index over which the summation is per-formed or the. To do this the inner indices need to be the same.

The index i is called a j free index. Trace of a product of matrices. The resulting language seems easy to use.

If you denote the entry on row column of an arbitrary matrix X by the definition of matrix multiplication says that. The Einstein summation convention is introduced. The next line is multiplication for E and C and then substitute the second line for e s entry.

Let A be the 4x4 matrix with on row column. That is show that ABC A BC for any matrices A B and C that are of the appropriate dimensions for matrix multiplication. The next line is multiplication in index notation with n o and p taking place of the dummy indices of i k and j respectively.

The abstract way to write a matrix multiplication with indices. The trace of a matrix is de ned to be the sum of its diagonal elements TrC X i C ii 12 We would like to prove that TrAB TrBA 13 2. Matrix multiplication The product of matrices AandBis deﬁned if thenumber of columns inAmatches the number ofrows inB.

C_ij A_ik B_kj Note that no dot is used in tensor notation The k in both factors automatically implies C_ij A_i1 B_1j A_i2 B_2j A_i3 B_3j which is the i th row of the first matrix multiplied by the j th column of the second matrix. V and index notation. 531 To get the element in the ith row and jth column of the product BA take the scalar product of the ith row-vector of Bwith the j-th column vector of A.

All the capabilities of matrix notation are retained and most carry over naturally to the n-way context. Begineqnarray C BA C_ij mathop _k12B_ ikA_kjquad text B_ikA_kj in the summation convention 531 endeqnarray. 712 Matrix Notation.

If one term has a free index i then to be consistent all terms must have it. The symbolic notation. LetA aik be anmnmatrix and bkj be annpmatrix.

For example one can multiply. Let B be the 4x4 matrix with on row column. The abstract way to write a matrix multiplication with indices.

If for example you want to compute C_23 then i2 and j3 and. TheproductABisdeﬁned to be thempmatrixC cij such that. Then substitute the first line in for d s entry.

We can use indices to write matrix multiplication in a more compact way. N-way generalization of matrix notation Summary The capabilities of matrix notation and algebra are generalized to n-way arrays. BikAkj in the summation convention.

The last equality in the quote above is simply row column of the matrix equality. B 2 ca 2. Let us look at scalar-vector multiplication in indicial notation.

B 1 ca 1. 133 Sigma Notation Sigma notation is used as a shorthand way of writing sums.

Applied Mechanics Of Solids A F Bower Appendix C Index Notation