Equivalent matrix?

[pivoxa15]

1. The problem statement, all variables and given/known data
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[i] module?

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





3. 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[i] module still stands. I know clearly what a Z[i] ring is.

[HallsofIvy]

Z[i] 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[i] module is refering to any module where the coefficients are from the ring of the Gaussian integers. Why use the word Gaussian?