document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=22654>Question 22654</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given two n x n matrices A and B where AB=BA how does one show that the <BR>\n" );
document.write( "determinant of (A^2 + B^2) >=0?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #11250 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=11250\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Given two n x n matrices A and B where AB=BA how does one show that the <BR>\n" );
document.write( "determinant of (A^2 + B^2) >=0? \r<BR>\n" );
document.write( "\n" );
document.write( " What level of linear algebra you are studying?\r<BR>\n" );
document.write( "\n" );
document.write( " It seems we have to use eigenvectors to prove it.\r<BR>\n" );
document.write( "\n" );
document.write( " AB=BA (commute) implies there is a basis of non-zero eigenvector say <BR>\n" );
document.write( " {vi | i=1,2,..n} in <BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%5En+ ALIGN=MIDDLE  ALT=\"F%5En+\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28R%5En%29 ALIGN=MIDDLE  ALT=\"%28R%5En%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that Avi = civi,Bvi = divi for some scalar (eigenvalues)<BR>\n" );
document.write( " ci,di for each i.\r<BR>\n" );
document.write( "\n" );
document.write( " Since for each i, we have (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2+%2B+B%5E2 ALIGN=MIDDLE  ALT=\"A%5E2+%2B+B%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)(vi) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2%28vi%29+%2B+B%5E2%28vi%29 ALIGN=MIDDLE  ALT=\"A%5E2%28vi%29+%2B+B%5E2%28vi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( " = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ci%5E2%2A+vi+%2B+di%5E2%2A+vi ALIGN=MIDDLE  ALT=\"ci%5E2%2A+vi+%2B+di%5E2%2A+vi\"   DISABLED_style= \"max-width:90%; height:auto;\" > = (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ci%5E2+%2B+di%5E2+ ALIGN=MIDDLE  ALT=\"ci%5E2+%2B+di%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >) vi.\r<BR>\n" );
document.write( "\n" );
document.write( "  Also, note that det(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2+%2B+B%5E2 ALIGN=MIDDLE  ALT=\"A%5E2+%2B+B%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) equals to the product of eigenvalues<BR>\n" );
document.write( " of the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2+%2B+B%5E2 ALIGN=MIDDLE  ALT=\"A%5E2+%2B+B%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " Hence, det(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2+%2B+B%5E2 ALIGN=MIDDLE  ALT=\"A%5E2+%2B+B%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PI%28ci%5E2+%2B+di%5E2+%29+%3E=0 ALIGN=MIDDLE  ALT=\"PI%28ci%5E2+%2B+di%5E2+%29+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (means product)\r<BR>\n" );
document.write( "\n" );
document.write( " Try to read carefully and understand the above proof.\r<BR>\n" );
document.write( "\n" );
document.write( " Good luck!\r<BR>\n" );
document.write( "\n" );
document.write( " Kenny\r<BR>\n" );
document.write( "\n" );
document.write( "   </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );