Skip to main content

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

Comments

leo said…
Nice post! There is a number of ways to generate nice LaTeX code directly inside a blog, without having to copy images from a PDF. See for example
http://sixthform.info/steve/wordpress/

Popular posts from this blog

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix.  Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume  x  where x is in the null space of A .  The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the null spa

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides:  A bit intuitive (for me) explanation of area of parallelogram  (to my site, external link) .