Matrices

Hoempa

Math Team
Apr 2010
2,780
361
For \(\displaystyle 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,
\(\displaystyle A = \frac{1}{2} \[\sqrt{5} - i \; \sqrt{5} + i \\ \sqrt{5} + i \; \sqrt{5} - i\]\), \(\displaystyle B = \[0 \; 0 \\ 0 \; 0\]\) works.
 

Hoempa

Math Team
Apr 2010
2,780
361
This might lead to a new family of 2x2 matrices such that \(\displaystyle A^2 + B^2 = \[2\;3\\3\;2\]\)

Where \(\displaystyle A^2 = \[w\;x\\x\;w\]\) and \(\displaystyle B^2 = \[y\;z\\z\;y\] = \[2-w\;3-x\\3-x\;2-w\]\)

Possible A's are

\(\displaystyle A = \[ a_1\;a_2 \\a_2 \;a_1 \]\)

Where \(\displaystyle a_2 = \pm \sqrt{\frac{w \pm \sqrt{w^2-x^2}}{2}}\) and

\(\displaystyle a_1 = \frac{x}{2 \cdot a_2}\)

B is constructed similarily.