For A^2 + B^2 = \[2 \; 3 \\ 3 \; 2\] I have a hard time finding A, B with both real non-zero entries. Complex entries enabled, A = \frac{1}{2} \[\sqrt{5} - i \; \sqrt{5} + i \\ \sqrt{5} + i \; \sqrt{5} - i\], B = \[0 \; 0 \\ 0 \; 0\] works.