Square matrices: $(A+B)^2=A^2+B^2$ - MathOverflow [closed]most recent 30 from http://mathoverflow.net2013-05-23T05:31:14Zhttp://mathoverflow.net/feeds/question/75767http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/75767/square-matrices-ab2a2b2Square matrices: $(A+B)^2=A^2+B^2$Pham Hung Quy2011-09-18T17:04:09Z2011-09-18T17:22:40Z
<p>If $a, b$ are two numbers such that $(a+b)^2 = a^2 + b^2$, then $a.b = 0$.</p>
<p>Is there a similar statement for square matrices.</p>
<p>"If $A, B$ are square matrices such that $(A+B)^2 = A^2 + B^2$, then $A.B = 0$."</p>
<p>Note that if $(A+B)^2 = A^2 + B^2$, then $AB = -BA$, hence $tr(AB) = 0$</p>
http://mathoverflow.net/questions/75767/square-matrices-ab2a2b2/75771#75771Answer by Chris Godsil for Square matrices: $(A+B)^2=A^2+B^2$Chris Godsil2011-09-18T17:22:40Z2011-09-18T17:22:40Z<p>If
<code>$$
A=\begin{pmatrix}
0&M\\ M&0\end{pmatrix},\qquad
B =\begin{pmatrix}-I&0\\ 0&I\end{pmatrix}
$$</code>
then $AB+BA=0$, so $A^2+B^2=(A+B)^2$ and neither $A$ nor $B$ is zero.</p>