Douglas Wilhelm Harder, M.Math., LEL

If you find this useful or would be interested in providing feedback and suggestions, please let me know at dwharder@gmail.com. See my other related pages on nice eigenvalue decompositions (eigendecompositions), integer matrices with integer eigenvalues, integer matrices with integer singular values, integer-normed vectors, integer-normed complex vectors, geometric sequences with nice norms.

When looking for examples in linear algebra courses, it is often preferable to have examples that have nice representations on the blackboard. Consequently, if the norm contains a surd, it becomes difficult to work with, and these examples are problematic in the class room.

To create a complex 2- or 3-dimensional vectors that have an integer norm, take a 3-, 4-, 5- or 6-dimensional real vector that has an integer norm and create complex numbers from those components, including a real or imaginary part equal to zero, as appropriate. For example, each of the following vectors has a 2-norm of 25:

     [ 9 12 20]          [0.36 0.48 0.8]
     [ 2  3  6 24]       [0.08 0.12 0.24 0.96]
     [ 2  3 12 12 18]    [0.08 0.12 0.48 0.48 0.72]
     [ 1  1  3 10 15 17] [0.04 0.04 0.12 0.4  0.6  0.68]

We can create the following complex vectors:

     [9-12j 20j]
     [2+3j -24+6j]
     [2+3j -12j 12+18j]
     [1+15j 17-1j 10+3j]

and each has a 2-norm of 25, so we can normalize these, as well:

     [0.36-0.48j 0.8j]
     [0.08+0.12j -0.96+0.24j]
     [0.08+0.12j -0.48j 0.48+0.72j]
     [0.04+0.6j 0.68-0.04j 0.4+0.12j]

Thus, each of these vectors is of unit length.