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?''
``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,


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.


[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


Reflection Line in 3 minutes

Reflection lines are lines observed on a surface from tube lights. Car designers employed this to see the surface quality. Usually, parallel fluorescent tubes are placed above the car model.

But this is quite common in Germany.

This is Berlin Hauptbahnhof (main station). The large part of building is transparent (glass). Steel frames are aligned regularly on the ceiling. These are not fluorescent tubes, but reflection line only requires regularly arranged lines. If there is something specular surface, we could see reflection lines.

Please look into the widows of this train. We could easily see that each window has a different quality. Most of the windows did not connects smoothly between windows.

Link to the explanation slides


[1] Mario Botsch, Mark Pauly, Leif Kobbelt, Pierre Alliez, Bruno Levy,
Stephan Bischo, Christian Roeossl, ``Geometric Modeling based on
Polygonal Meshes,'' EG2008 Tutorial, 2008

[2] Takashi Kanai, Yusuke Yasui, ``High-quality Display of Subdivision
Surfaces on GPU,'' IPSJ 47(2), pp.647-655, 2006 [in Japanese]