Equivalent Matrix in Z[i] Module: How to Transform Vectors?

[pivoxa15]

Homework Statement

How do you turn the transformation matrix

1+i 3
2-i 5i

into a diagonal matrix when it is transforming vectors in the Z module?

What is the Z module? Normally one states R module M. where R is the coefficient one multiplies to the elements in the module.

The Attempt at a Solution

I have tried but can't seem to reduce to diagonal form as I can only multiply by integer numbers?

 

[pivoxa15]

I have just worked out the diagonal matrix. It is

1 0
0 8+11i

However my question of what is the Z module still stands. I know clearly what a Z ring is.

 

[HallsofIvy]

Z is the same whether you think of it as a ring or a module- the "Gaussian integers"- all numbers of the form a+ bi where a and b are integers and i is, of course, the imaginary unit.

 

[pivoxa15]

I think that the Z module is referring to any module where the coefficients are from the ring of the Gaussian integers. Why use the word Gaussian?