Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
No subject limitations
Search only in selected subjects
    Calculus or Analysis
    Number Theory
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Advanced level:

Really Complex Matrices

If you know how to multiply 2x2 matrices, and know about complex numbers, then you'll enjoy this connection. Any complex number (a+bi) can be represented by a real 2x2 matrix in the following way! Let the 2x2 matrix

[  a b ]
[ -b a ]

correspond to (a+bi). Addition of complex numbers then corresponds to addition of the corresponding 2x2 matrices. So does multiplication! Observe if you take this product:

[  a b ] [ c d ]
[ -b a ] [ -d c ]

you get

 [  (ac-bd) (ad+bc) ]
[ -(ad+bc) (ac-bd) ]

which is precisely what you would get if you multiplied (a+bi) and (c+di) and then converted to a 2x2 matrix!

Presentation Suggestions:
Let students do the multiplication, or maybe have done it already for homework before you present this fun fact. As a follow up Fun Fact, note that taking determinants of these matrices produce Products of Sums of Two Squares.

The Math Behind the Fact:
The reason this works is because complex multiplication can be viewed as a linear transformation on the 2-dimensional plane. In linear algebra, you learn that every linear transformation can be represented as matrix multiplication by a suitable matrix.

How to Cite this Page:
Su, Francis E., et al. "Really Complex Matrices." Math Fun Facts. <>.

Keywords:    complex numbers, linear algebra
Subjects:    algebra
Level:    Advanced
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!