Question: Smith Normal Form of Complex Polynomials Matrix


I have a question in factoring a matrix of complex polynomials into its Smith Normal Form. It seems that Maple is not giving me correct answers.

I tried a simple matrix:

A:=Matrix([[I*z-1, z+I], [z+I, z+I]]);

Maple gives me:

SmithForm( A, z, method='rational'  ,output=['S'] );

which is:

Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -4*z^2-(3*I)*z^3+z^4+1+(3*I)*z})

Since the gcd of the 4 elements is (z+i), it seems the Smith Normal Form should not be in this form. Also, I got errors in solving the other 2 matrices:

SmithForm( A, z, method='rational'  ,output=['U'] );
Error, (in convert/radical) numeric exception: division by zero


How to make it work for complex polynomials?

Sorry that I am not an expert in Algebra~

Thanks a lot!


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