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Unformatted text preview: A mystery of transpose of matrix multiplication Why ( AB ) T = B T A T ? Yamauchi, Hitoshi Nichiyou kenkyu aikouka Berlin, Germany 2009-5-24(Sun) Contents 1 Introduction 1 2 Dot product of vectors 1 3 Matrix Multiplication 3 4 Transpose of Matrix Multiplication 4 Abstract Why does the transpose of matrix multiplica- tion is ( AB ) T = B T A T ? I was told this is the rule when I was a university student. But, there must be something we could understand this. First, I would like to show you that the relation- ship of dot product of vectors and its transpose is bracketleftbig u T v bracketrightbig T = vu T . Then I will point out to you that a matrix is a representation of transform rather a representation of simultaneous equa- tions. This point of view gives us that a matrix multiplication includes dot products. Combin- ing these two point of views and one more, a vector is a special case of a matrix, we could understand the first equation. 1 Introduction This article talks about the transpose of matrix multiplication ( AB ) T = B T A T . (1) When I saw this relationship, I wonder why this happens. For me, transpose is an operator, it looks like this is a special distribution law. If multiply something (for instance, -1), ( − 1) · ( a + b ) = ( − 1 · a ) + ( − 1 · b ) . There is no place exchange of a and b , it does not become b a . But in the case of transpose, ( AB ) T = B T A T bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright The order AB becomes BA . . (2) Why this happens? If I compare the each of element, I could see this should be, however, I feel something magical and could not feel to understand it. Recently, I read a book and felt a bit better. So I would like to introduce the explanation. Before we start to talk about the matrix mul- tiplication, I would like to start from dot prod- uct of vectors since vector is a special case of ma- trix. Then we will generalize back this idea to matrix multiplication. Because a simpler form is usually easier to understand, we will start a simple one and then go further. 2 Dot product of vectors Let’s think about two vectors u , v . Most of the linear algebra textbooks omit the elements of vector since it is too cumbersome, however, I will put the elements here. When we wrote ele- ments like u = u 1 u 2 . . . u n (3) as a general vector, this is also cumbersome. So, I will start with three dimensional vector. Then we could extend this to general dimensional vec- tors. Here the main actor of this story is trans- pose T , this is an operator to exchange the row 1 and column of a matrix or a vector....
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