The cube roots of

We all know that the cube root of 1 is 1. But what is the cube root of the 3 by 3 identity matrix?

Consider the following matrix:

> A := matrix([[-1,-6,-4],[-4,4,1],[7,-9,-3]]);

> evalm(A^2),evalm(A^3),evalm(A^4),evalm(A^5),evalm(A^6);

We can see that A cubed is the 3 by 3 identity matrix, so in some sense A is the cube root of the 3 by 3 identity matrix.

Do all 3 by 3 matrices have this property?

> B:=matrix(3,3,rand(-10..10));

> evalm(B^2),evalm(B^3),evalm(B^4),evalm(B^5),evalm(B^6);

Obviously not!

Is A the unique cube root of the 3 by 3 identity matrix? No, in fact there are an infinite number of them!

Think in terms of linear transformations. A is a linear transformation which takes us back to where we started from after doing the transformation 3 times. There are an infinite number of such transformations. Below is an example of how to find one, starting with a random non-singular 3 by 3 matrix,

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Finding a cube root of starting from a random 3 by 3 matrix: