2009-05-16

A mystery of transpose of matrix multiplication -- Why (AB)^T = B^T A^T?

At one Wednesday night party (meeting), there was a conversation:

``... I met the woman at that party. She was looking for her job...''
``What does she look like? Pretty? Positive? Tall?''
``Positive.''
``I love a positive woman.''

A few minutes later at the same party,

``... I found this matrix in that paper. It was constructed by...''
``What does the matrix look like? Symmetric? Positive definite? How large?''
``Positive definite.''
``I love a positive definite matrix.''

As you see, some of my friends like a positive definite quadratic form matrix. It is also interesting to think about the relationship of this matrix with transpose. But first I would like to talk about transpose only.

One of my friend wrote an article ``A mystery of transpose of matrix multiplication -- Why (AB)^T = B^T A^T?'' By the way, it is not comfortable to write a mathematical blog since it is hard to write equations. I have a permission to copy it here, but I will just link the article. The following is the abstract of the article.

A transpose of matrix multiplication is

Eq (1)

Sometimes we just say this is a rule. But, I would like to think about this more. First, I would like to think about vector since vector is a special case of matrix and it is simpler than matrix. From a transpose of dot product of vector,

Eq(2)

is shown as a natural rule. This is well explained in the Farin and Hansford book [1]. I added some additional details with the book's explanation. I recommend this book.

Matrix multiplication includes dot product if you can see that a matrix is also a representation of transform. This matrix's aspect is emphasized and well explained in the Sugihara's book [2]. I recommend this book, an unfortunate for non Japanese readers, the book is written in Japanese.

If you sum these two ideas, I hope you will see the meaning of Equation 1 better.




References

[1] Gerald Farin and Dianne Hansford, Practical Linear Algebra; A Geometry Toolbox, A K Peters, Ltd., 2005, ISBN: 1-56881-234-5

[2] Koukichi Sugihara, Mathematical Theory of Graphics (Gurafikusu no Suuri), Kyouritu-shuppan, 1995, ISBN: 4-320-02663-2 C3341

1 comment:

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