Cholesky Matrix

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Cholesky Matrix</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="95%" align="left" valign="top"><p class="word_list"> <a href="#Cholesky algorithm">Cholesky algorithm</a><p class="word_list"> <a href="#Cholesky factorization">Cholesky factorization</a><p class="word_list"> <a href="#Cholesky matrix">Cholesky matrix</a><p class="word_list"> <a href="#indefinite">indefinite matrix</a><p class="word_list"> <a href="#negative definite">negative definite matrix</a><p class="word_list"> <a href="#negative semidefinite">negative semidefinte matrix</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table5"> <tr> <td align="right" width="9" height="14"></td> <td width="300" height="14"> <font size="2"> </font><font size="4"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body">If we think of matrices as multi-dimensional generalizations of numbers, we may draw useful analogies between numbers and matrices. Not least of these is an analogy between positive numbers and positive definite matrices. </p> <p class="body">Just as we can take square roots of positive numbers, so can we take “square roots” of positive definite matrices. </p> <p class="body"><span class="header_section_green">Positive Definite Matrices <br> </span>A symmetric matrix <i><b>x</b></i> is said to be: </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/positive_definite_matrix.htm">positive definite</a> if <img border="0" src="../formulas/cholesky_factorization_bxb.gif" align="absbottom" width="29" height="16"> > 0 for all row vectors <i><b>b</b></i> <img border="0" src="../symbols/not_equal.gif" align="absbottom" width="9" height="15"> <b>0</b>; </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/positive_definite_matrix.htm">positive semidefinite</a> if <img border="0" src="../formulas/cholesky_factorization_bxb.gif" align="absbottom" width="29" height="16"> <img border="0" src="../symbols/greater_than_or_equal.gif" align="absbottom" width="9" height="14"> 0 for all row vectors <i><b>b</b></i>; </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><b><a name="negative definite">negative definite</a></b> if <img border="0" src="../formulas/cholesky_factorization_bxb.gif" align="absbottom" width="29" height="16"> < 0 for all row vectors <i><b>b</b></i> <img border="0" src="../symbols/not_equal.gif" align="absbottom" width="9" height="15"> <b>0</b>; </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><b><a name="negative semidefinite">negative semidefinite</a></b> if <img border="0" src="../formulas/cholesky_factorization_bxb.gif" align="absbottom" width="29" height="16"> <img border="0" src="../symbols/less_than_or_equal.gif" align="absbottom" width="9" height="14"> 0 for all row vectors <i><b>b</b></i>; </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><b><a name="indefinite">indefinite</a></b> if none of the above hold. </p> <p class="body">These definitions may seem abstruse, but they lead to an intuitively appealing result. A symmetric matrix <i><b>x</b></i> is: </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">positive definite if all its <a href="../articles/eigenvalue.htm">eigenvalues</a> are <a href="../articles/set.htm">real</a> and positive; </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">positive semidefinite if all its eigenvalues are real and nonnegative; </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">negative definite if all its eigenvalues are real and negative; </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">negative semidefinite if all its eigenvalues are real and nonpositive; </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">indefinite if none of the above hold. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table6"> <tr> <td align="right" width="345" colspan="3" height="14"></td> </tr> <tr> <td align="right" width="9" rowspan="2"> </td> <td width="168" height="15"> <p class="ad_legend"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a>  </td> <td width="168" height="15" align="right"> <p class="ad_legend"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Advertise on this site</font></a></td> </tr> <tr> <td width="336" colspan="2">  <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=84506;type=iframe" allowtransparency="true" background-color="transparent" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I1"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=84506;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=84506"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=84506;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>  </td> </tr> </table> <p class="body">It is useful to think of positive definite matrices as analogous to positive numbers and positive semidefinite matrices as analogous to nonnegative numbers. The essential difference between semidefinite matrices and their definite analogues is that the former can be singular whereas the latter cannot. This follows because a matrix is singular if and only if it has a 0 eigenvalue. </p> <p class="body"><span class="header_section_green">Matrix “Square Roots” <br> </span>Nonnegative numbers have real square roots. Negative numbers do not. An analogous result holds for matrices. Any positive semidefinite matrix <i><b>h</b></i> can be factored in the form <img border="0" src="../formulas/cholesky_factorization_h_kk.gif" align="absbottom" width="48" height="16"> for some real square matrix <i><b>k</b></i>, which we may think of as a matrix square root of <i><b>h</b></i>. The matrix <i><b>k</b></i> is not unique, so multiple factorizations of a given matrix <i><b>h</b></i> are possible. This is analogous to the fact that square roots of positive numbers are not unique either. If <i><b>h</b></i> is nonsingular (positive definite), <i><b>k</b></i> will be nonsingular. If <i><b>h</b></i> is singular, <i><b>k</b></i> will be singular. </p> <p class="body"><span class="header_section_green">Cholesky Factorization </span><br> A particularly easy factorization <img border="0" src="../formulas/cholesky_factorization_h_kk.gif" align="absbottom" width="48" height="16"> to perform is one known as the <b><a name="Cholesky factorization"> Cholesky factorization</a></b>. Any positive semidefinite matrix has a factorization of the form <img border="0" src="../formulas/cholesky_factorization_h+gg.gif" align="absbottom" width="47" height="16"> where <i><b>g</b></i> is a lower triangular matrix. Solving for <i><b>g</b></i> is straightforward. Suppose we wish to factor the positive definite matrix </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[1].gif" width="100" height="72"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">A Cholesky factorization takes the form </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table8"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[2].gif" width="374" height="75"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">By inspection, <img border="0" src="../formulas/cholesky_factorization_fragment_1.gif" align="absbottom" width="177" height="20"> Also by inspection, <img border="0" src="../formulas/cholesky_factorization_fragment_2.gif" align="absbottom" width="84" height="15"> Since we already have <img border="0" src="../formulas/cholesky_factorization_fragment_3.gif" align="absbottom" width="50" height="15">, we conclude <img border="0" src="../formulas/cholesky_factorization_fragment_4.gif" align="absbottom" width="56" height="15">. Proceeding in this manner, we obtain a matrix <i><b>g</b></i> in 6 steps: </p> <p class="body"> <img border="0" src="../formulas/cholesky_factorization_bullets.gif" width="304" height="236"></p> <p class="body">Our <b><a name="Cholesky matrix">Cholesky matrix</a></b> is </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[3].gif" width="142" height="82"></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">The above example illustrates a <b> <a name="Cholesky algorithm">Cholesky algorithm</a></b>, which generalizes for higher dimensional matrices. Our algorithm entails two types of calculations: </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table10"> <tr> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> <p class="body">1. Calculating diagonal elements <img border="0" src="../formulas/cholesky_factorization_g_ii.gif" align="absbottom" width="18" height="11"> (steps 1, 4, and 6) entails taking a square root. </p> <p class="body">2. Calculating off-diagonal elements <img border="0" src="../formulas/cholesky_factorization_g_ij.gif" align="absbottom" width="19" height="11">, <i>i</i> > <i>j</i>, (steps 2, 3, and 5) entails dividing some number by the last-calculated diagonal element. </p> <p class="body">For a positive definite matrix <i><b>h</b></i>, all diagonal elements <img border="0" src="../formulas/cholesky_factorization_g_ii.gif" align="absbottom" width="18" height="11"> will be nonzero. Solving for each entails taking the square root of a nonnegative number. We may take either the positive or negative root. Standard practice is to take only positive roots. Defined in this manner, the Cholesky matrix of a positive definite matrix is unique. </p> <p class="body">The same algorithm applies for singular positive semidefinite matrices <i><b>h</b></i>, but the result is not generally called a Cholesky matrix. This is just an issue of terminology. When the algorithm is applied to the singular <i><b>h</b></i>, at least one diagonal elements <img border="0" src="../formulas/cholesky_factorization_g_ii.gif" align="absbottom" width="18" height="11"> equals 0. If only the last diagonal element <img border="0" src="../formulas/cholesky_factorization_g_nn.gif" align="absbottom" width="24" height="12"> equals 0, we can obtain <i><b>g</b></i> as we did in our example. If some other diagonal element <img border="0" src="../formulas/cholesky_factorization_g_ii.gif" align="absbottom" width="18" height="11"> equals 0, off-diagonal element <img border="0" src="../formulas/cholesky_factorization_g_i1i.gif" align="absbottom" width="31" height="12"> will be indeterminate. We can set such indeterminate values equal to any value within an interval [–<i>a</i>, <i>a</i>], for some a <img border="0" src="../symbols/greater_than_or_equal.gif" align="absbottom" width="9" height="14"> 0. </p> <p class="body">Consider the matrix </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table11"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[4].gif" width="109" height="62"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">Performing the first four steps of our algorithm above, we obtain </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table12"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[5].gif" width="367" height="71"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">In the fifth step, we multiply the second row of <i><b>g</b></i> by the third column of <i><b>g'</b></i> to obtain </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table13"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[6].gif" width="155" height="20"></td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">We already know <img border="0" src="../formulas/cholesky_factorization_fragment_5.gif" align="absbottom" width="203" height="15">, so we have </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table14"> <tr> <td width="100%" align="center" rowspan="2"> <img border="0" src="../formulas/cholesky_factorization__[7]_[8].gif" width="159" height="76"></td> <td width="5%" align="right">[7]</td> </tr> <tr> <td width="5%" align="right">[8]</td> </tr> </table> <p class="body">which provides us with no means of determining <img border="0" src="../formulas/cholesky_factorization_g32.gif" align="absbottom" width="24" height="12">. It is indeterminate, so we set it equal to a variable <i>x</i> and proceed with the algorithm. We obtain </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table15"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[9].gif" width="194" height="89"></td> <td width="5%" align="right">[9]</td> </tr> </table> <p class="body">For the element <img border="0" src="../formulas/cholesky_factorization_g_33.gif" align="absbottom" width="24" height="12"> to be real, we can set <i>x</i> equal to any value in the interval [–3,3]. The interval of acceptable values for indeterminate components will vary, but it will always include 0. For this reason, it is standard practice to set all indeterminate values equal to 0. With this selection, we obtain </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table16"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[10].gif" width="114" height="64"></td> <td width="5%" align="right">[10]</td> </tr> </table> <p class="body">We can leave <i><b>g</b></i> in this form, or we can delete the second column, which contains only 0’s. The resulting 3<img border="0" src="../symbols/cross.gif" align="absbottom" width="11" height="13">2 matrix provides a valid factorization of <i><b>h</b></i> since </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table17"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[11].gif" width="415" height="65"></td> <td width="5%" align="right">[11]</td> </tr> </table> <p class="body">If a matrix <i><b>h</b></i> is not positive semidefinite, our Cholesky algorithm will, at some point, attempt to take a square root of a negative number and fail. Accordingly, the Cholesky algorithm is a means of testing if a matrix is positive semidefinite. </p> <p class="body"><span class="header_section_green">Computational Issues </span><br> In exact arithmetic, the Cholesky algorithm will run to completion with all diagonal elements <img border="0" src="../formulas/cholesky_factorization_g_ii.gif" align="absbottom" width="18" height="11"> > 0 if and only if the matrix <i><b>h</b></i> is positive definite. It will run to completion with all diagonal elements <img border="0" src="../formulas/cholesky_factorization_g_ii.gif" align="absbottom" width="18" height="11"> <img border="0" src="../symbols/greater_than_or_equal.gif" align="absbottom" width="9" height="14"> 0 and at least one diagonal element <img border="0" src="../formulas/cholesky_factorization_g_ii.gif" align="absbottom" width="18" height="11"> = 0 if and only if the matrix <i><b>h</b></i> is singular positive semidefinite. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table18"> <tr> <td align="right" width="9"> </td> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body">Things are more complicated if arithmetic is performed with rounding, as is done on a computer. Off-diagonal elements are obtained by dividing by diagonal elements. If a diagonal element is close to 0, any roundoff error may be magnified in such a division. For example, if a diagonal element should be .00000001, but roundoff error causes it to be calculated as .00000002, division by this number will yield an off-diagonal element that is half of what it should be. </p> <p class="body">An algorithm is said to be unstable if roundoff error can be magnified in this way or if it can cause the algorithm to fail. The Cholesky algorithm is unstable for singular positive semidefinite matrices <i><b>h</b></i>. It is also unstable for positive definite matrices <i><b> h</b></i> that have one or more eigenvalues close to 0. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table19"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black"><a name="Exercises">Exercises</a></td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table20"> <tr> <td width="100%"> <p class="body" style="margin-left: 10; margin-top: 8; margin-bottom: 8"> <img border="0" src="../images/bullet_exercise.gif" align="absbottom" width="19" height="17">Identify all factorizations of the following matrices that are obtainable with the Cholesky algorithm. Take only positive square roots when selecting nonzero diagonal elements <img border="0" src="../formulas/cholesky_factorization_g_ii.gif" align="absbottom" width="18" height="11">. In each case, is the original matrix positive definite, singular positive semidefinite, or neither of these?<p style="margin-left: 10">a. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table21"> <tr> <td width="98%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[e1].gif" width="105" height="91"></td> <td width="2%">[e1]</td> </tr> </table> <p style="margin-left: 10">b. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table22"> <tr> <td width="98%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[e2].gif" width="108" height="91"></td> <td width="2%">[e2]</td> </tr> </table> <p style="margin-left: 10">c. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table23"> <tr> <td width="98%" align="center"> <img border="0" src="../formulas/cholesky_factorization_[e3].gif" width="100" height="91"></td> <td width="2%">[e3]</td> </tr> </table> <p class="body" style="margin-left: 10; 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