Matrix Multiplication Sum Notation
Matrix-Column Vector Product Now lets take matrix-column vector multiplication Ax b. 33 Representing Summation Using Matrix Notation Consider the sum.
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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.
Matrix multiplication sum notation. For a set of n a values this becomes Xi n i 1 ai A3 which is usually abbreviated to either Xn i 1 ai A4 Notation Matrices and Matrix Mathematics 375. In my experience what causes the most confusion is not the subscripts indicating the component but the meaning of the summation notation. A 1 3 5 7 B 2 4 6 8.
C_ik A_ij Bjk In matrix representation the first index always represents rows and the second always columns. We show how to use index notation and sum over row and column indices to perform matrix multiplication. Indicating that summation of asetofavaluesshouldbecarriedouton allthe elements from a 1 to a 6.
This is illustrated below by positioning B above and to the right with A down and to the left to highlight that we. We can do addition and subtraction. Sigma summation and Pi product notation are used in mathematics to indicate repeated addition or multiplication.
This representation works well for order 1 and 2 tensors but begins to be ugly for bigger orders. We can use indices to write matrix multiplication in a more compact way. AB BA the other rule in addition of algebra are also valid in addition of matrices.
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. 3 7 7 7 5 2 6 6 6 4 x 1 x 2 x 3. Pi notation provides a compact way to represent many products.
Sigma notation provides a compact way to represent many sums and is used extensively when working with Arithmeticor Geometric Series. The inner indices run from 1 to n so you can introduce a summation index k and write this sum compactly using summation notation. Let mathbfxx_1ldotsx_nprime be an ntimes1 vector and mathbf11ldots1prime be an ntimes1 vector of ones.
Multiplication of a matrix by a number is called scalar multiplication. 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. The important thing is the matrices must have same dimension.
The product of two matrices and is defined as 1 where is summed over for all possible values of and and the notation above uses the Einstein summation convention. So the correct notation is. C i j k 1 n a i k b k j The formule above thus gives you the element on position i j in the product matrix C A B and therefore completely defines C by letting i 1 m and j 1 p.
Or written in a more compact notation ab X i a ib i. For example if there are matrix A and B then. 3 7 7 7 5 2 6 6 6 4 b 1 b 2 b 3.
133 Sigma Notation Sigma notation is used as a shorthand way of writing sums. 2 where the means that we sum over all values of i. Basic Algorithms and Notation 2.
Structure and Efficiency 3. It is just adding the numberelement of matrix one by one as each position. Lets cover the kinds of math operations we can do.
The Einstein summation convention is introduced. The implied summation over repeated indices without the presence of an explicit sum sign is called Einstein summation and is commonly used in both matrix and tensor analysis. 3 7 7 7 5 3.
In class I define the matrix product by A B i j k 1 q A i k B k j where A is p q and B is q r. Matrix addition and subtraction is the same as number addition. B 20 28 52 76.
Matrix Addition and Subtraction. 2 6 6 6 4 A 11 A 12 A 23 A 21 A 22 A 23 A 31 A 32 A 33. Say we have two tables A and B.
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