Based on what condition a Diagonalized of a matrix could be all integer numbers?

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I need to get a Diagonalized of a matrix. My given matrix is 4*4 strictly Dominant Diagonal matrix. something like this: \begin{array}{l}4&2&1&0\\1&5&2&1\\2&0&6&3\\1&1&0&3\end{array} it seems the result of Diagonalized of matrix ( if it is possible) would be in form of A= S * D * S_inverse. So now I really need to have the D matrix as all entries are integers not fractional. is there any way or any condition that A as strictly Dominant Diagonal matrix can have D as diagonal matrix with all integers? I used this website for matrix calculation, it is hard to just try infinite numbers to get finally your desired output.
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EDITED strictly Dominant Diagonal matrix : a matrix which items on diagonal is strictly greater than absolute sum of all items in that row, means a_ii > Sum ( | a_ij | ) for all j, i != j as you see in my example 4 > 2 +1 + 0 and 5 > 1 + 2 + 1 and so on