document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194124>Question 1194124</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, i don't really know what to do here.\r<BR>\n" );
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document.write( "The linear operator A is called the idempotent or the projection operator if A^2 = A. What is the canonical operator of the idempotent operator Jordan?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848534 by </B><A HREF=/tutors/aboutme.mpl?userid=parmen><B>parmen(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=parmen><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=848534\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **Key Properties of Idempotent Operators:**\r<BR>\n" );
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document.write( "* **Definition:** An operator A is idempotent if A² = A.<BR>\n" );
document.write( "* **Eigenvalues:** The only possible eigenvalues of an idempotent operator are 0 and 1.<BR>\n" );
document.write( "* **Diagonalizability:** Idempotent operators are always diagonalizable.\r<BR>\n" );
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document.write( "**Canonical Form of an Idempotent Operator:**\r<BR>\n" );
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document.write( "The canonical form of an idempotent operator is a diagonal matrix with only 0s and 1s on the diagonal. \r<BR>\n" );
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document.write( "**Explanation:**\r<BR>\n" );
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document.write( "* **Diagonalization:** Since idempotent operators are diagonalizable, there exists an invertible matrix P such that:<BR>\n" );
document.write( "   P⁻¹AP = D <BR>\n" );
document.write( "   where D is a diagonal matrix.\r<BR>\n" );
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document.write( "* **Eigenvalues and Diagonal:**<BR>\n" );
document.write( "   - The diagonal entries of D represent the eigenvalues of A.<BR>\n" );
document.write( "   - Since the only eigenvalues of an idempotent operator are 0 and 1, the diagonal of D will consist of only 0s and 1s.\r<BR>\n" );
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document.write( "**Example:**\r<BR>\n" );
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document.write( "Let A be an idempotent operator. Its canonical form (Jordan form) would be:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "D = <BR>\n" );
document.write( "| 1 0 0 0 |<BR>\n" );
document.write( "| 0 1 0 0 |<BR>\n" );
document.write( "| 0 0 0 0 |<BR>\n" );
document.write( "| 0 0 0 0 | <BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "This represents a 4x4 matrix with two eigenvalues: 1 (with multiplicity 2) and 0 (with multiplicity 2).\r<BR>\n" );
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document.write( "**In Summary:**\r<BR>\n" );
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document.write( "The canonical form of an idempotent operator is a diagonal matrix with only 0s and 1s on the diagonal, reflecting its eigenvalues and its diagonalizability.<BR>\n" );
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