document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=29838>Question 29838</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I dont understand these Partician Matrices and how to break it up.\r<BR>\n" );
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document.write( "Compute the following matrix<BR>\n" );
document.write( "1 0 0 0<BR>\n" );
document.write( "0 1 0 0 <BR>\n" );
document.write( "0 1 1 0<BR>\n" );
document.write( "0 -1 0 1 \r<BR>\n" );
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document.write( "The entire matrix is to the exponent 2006\r<BR>\n" );
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document.write( "Thank you so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #16590 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=16590\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Some facts that you have to know:<BR>\n" );
document.write( " 1) The Caley Hamilton Theorem:<BR>\n" );
document.write( " If f(x) is the characteristic polynomial for a given matrix A,<BR>\n" );
document.write( " then f(A)= 0. [Note deg of f(x) = size A = n)<BR>\n" );
document.write( " Now A is 4 x4 matrix, deg char poly of A = 4, so any high power <BR>\n" );
document.write( " of A can be reduced to at most of exponeent 3 by division.\r<BR>\n" );
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document.write( " 2) To find the minnimal polynomial m(x) for A, i.e. the lowest<BR>\n" );
document.write( " deg (real heer)poy. such that m(A)= 0. Also, m(x) must divide<BR>\n" );
document.write( " char for A and contains all eigenvalues for A as roots.<BR>\n" );
document.write( " 3) A is diagonalizable (in C)iff the min. ploy. for A is<BR>\n" );
document.write( " the product of distinct linear factors.\r<BR>\n" );
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document.write( " Now, clearly,this given matrix has 1 as the unique eigenvalue.<BR>\n" );
document.write( " (actually, such matrix ialled unipotent, i.e. (A-I)^k = 0 for some I)<BR>\n" );
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document.write( " to find A^2006 , we have to get its min. poly. first.<BR>\n" );
document.write( " <BR>\n" );
document.write( " Since A - I =  <BR>\n" );
document.write( " [0, 0, 0, 0<BR>\n" );
document.write( "  0, 0, 0, 0 <BR>\n" );
document.write( "  0, 1, 0, 0<BR>\n" );
document.write( "  0, -1, 0, 0]<BR>\n" );
document.write( " by dirct computation (A-I)^2 = 0\r<BR>\n" );
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document.write( " So, m_A(x) = (x-1)^2 (min. poly.)<BR>\n" );
document.write( " Then try to find the remainder of x^2006 divided by (x-1)^2:<BR>\n" );
document.write( " Since x^2006 -1 = (x^2 -1) q(x) <BR>\n" );
document.write( " [ or sove x^= (x^2-1) q(x) + ax + b for a & b. Don't use brutal force <BR>\n" );
document.write( " of long division]<BR>\n" );
document.write( " So,  x^2006 = 1 mod (x^2 -1) <BR>\n" );
document.write( " Therefore A^2006 = I.\r<BR>\n" );
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document.write( " Make sure you understand about each step.<BR>\n" );
document.write( " By the way, [since nullitt(A-I) = 3 & m(x)=(x-1)^2] <BR>\n" );
document.write( " the Jordan Canonical Form for A is (i.e A is similar to)<BR>\n" );
document.write( " (1 0  0 0)<BR>\n" );
document.write( " (1 1  0 0)<BR>\n" );
document.write( " (0 0 1 0)<BR>\n" );
document.write( " (0 0 0 1)\r<BR>\n" );
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document.write( " Kenny\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( "  </SPAN>\n" );
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