1

Can somebody help me with the Smith normal form of this matrix? I know what it should be, but I get stuck at some point. Can you show how to take it from the point I'm stuck?

This is the matrix:

original matrix http://imagizer.imageshack.us/v2/150x100q90/661/KhsK4K.png

And this is the point where I'm stuck:

second matrix http://imagizer.imageshack.us/v2/150x100q90/909/iOfRTG.png

Thank you very much for your help!

Andrei Stirb
  • 13

1 Answers1

1

Expanding out the brackets, we have

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & x^{2}-8x+16 & 0 \\ 0 & 0 & 0 & x^{2}-3x+2 \end{pmatrix}

Now add row 4 to row 3 to get

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & x^{2}-8x+16 & x^{2}-3x+2 \\ 0 & 0 & 0 & x^{2}-3x+2 \end{pmatrix}

Subtract column 3 from column 4 to get an entry of lower degree:

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & x^{2}-8x+16 & 5x-14 \\ 0 & 0 & 0 & x^{2}-3x+2 \end{pmatrix}

Swap columns 3 and 4:

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5x-14 & x^{2}-8x+16 \\ 0 & 0 & x^{2}-3x+2 & 0 \end{pmatrix}

I think if you got this far, you should be able to figure out the rest.

A Kubiesa
  • 698