{"id":3701,"date":"2018-08-20T10:55:45","date_gmt":"2018-08-20T15:55:45","guid":{"rendered":"https:\/\/blogs.mathworks.com\/cleve\/?p=3701"},"modified":"2018-08-30T14:33:43","modified_gmt":"2018-08-30T19:33:43","slug":"reviving-wilsons-matrix","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/cleve\/2018\/08\/20\/reviving-wilsons-matrix\/","title":{"rendered":"Reviving Wilson&#8217;s Matrix"},"content":{"rendered":"<div class=\"content\"><!--introduction--><p>I probably first saw this matrix in 1960 in John Todd's class at Caltech. But I forgot about it until Tahar Loulou jogged my memory with a comment following <a href=\"https:\/\/blogs.mathworks.com\/cleve\/2018\/05\/28\/the-jordan-canonical-form-just-doesnt-compute\">my blog post in late May<\/a>. It deserves a place in our gallery of interesting matrices.<\/p><!--\/introduction--><h3>Contents<\/h3><div><ul><li><a href=\"#35987f7a-3b78-4662-99fc-3e903cbe9977\">T. S. Wilson<\/a><\/li><li><a href=\"#117510f9-c60f-4bd6-b6d6-03aa85c0c393\">T. S. Wilson's Matrix<\/a><\/li><li><a href=\"#8e7b2cf8-a032-4398-b454-07a2044d663e\">Determinant and Inverse<\/a><\/li><li><a href=\"#594e4474-66f7-4e9c-a5ee-4136c7f71922\">Cholesky<\/a><\/li><li><a href=\"#ba455ab8-3cea-4ae9-8717-af0978569546\">Eigenvalues<\/a><\/li><li><a href=\"#0c105ada-0a76-445e-b318-177105a15057\">Condition<\/a><\/li><li><a href=\"#9461b772-cf36-47af-9f4c-cd4abb09d6b6\">An Experiment<\/a><\/li><li><a href=\"#8cf06111-dff0-4ec8-8abd-ef7364be477e\">Thanks<\/a><\/li><li><a href=\"#086a6209-ef3f-4153-8a00-8eb297f35f5c\">References<\/a><\/li><\/ul><\/div><h4>T. S. Wilson<a name=\"35987f7a-3b78-4662-99fc-3e903cbe9977\"><\/a><\/h4><p>I'm afraid that I don't know very much about T. S. Wilson himself. The only paper we could find on the Internet is the joint paper about pendulum dampers listed in the references.  This paper, which does not mention the matrix, was published in 1940.  The authors were from a British company called D. Napier &amp; Son Ltd. The earliest appearance of the matrix that we know of  is the 1946 paper by J. Morris referenced below.  It is possible that by this time Wilson and Morris were colleagues at the Royal Aircraft Establishment in Farnborough in the UK.<\/p><p>I think that Wilson and Todd must have known each other. Todd acknowledges Wilson in several papers and books.<\/p><p>One of the references we used in Todd's class is the booklet written by Marvin Marcus also referenced below.  The booklet has a short section collecting some \"particular\" matrices.  A_13 is our old friend the Hilbert matrix and A_14 is Wilson's matrix.<\/p><h4>T. S. Wilson's Matrix<a name=\"117510f9-c60f-4bd6-b6d6-03aa85c0c393\"><\/a><\/h4><p>Here is the matrix. Let's use the Live Editor&reg; and the Symbolic Math Toolbox&reg;.<\/p><pre class=\"codeinput\">    W = sym([5 7 6 5; 7 10 8 7; 6 8 10 9; 5 7 9 10]);\r\n<\/pre><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/W.png\" alt=\"\"> <\/p><h4>Determinant and Inverse<a name=\"8e7b2cf8-a032-4398-b454-07a2044d663e\"><\/a><\/h4><p>The first noteworthy property is the determinant.  The determinant has to be an integer because all the matrix elements are integers.<\/p><pre class=\"codeinput\">    d = det(W);\r\n<\/pre><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/det.png\" alt=\"\"> <\/p><p>This is the smallest possible determinant for a nonsingular matrix with integer entries.<\/p><p>Since the determinant is one, the inverse must also have integer entries.<\/p><pre class=\"codeinput\">    Winv = round(inv(W));\r\n<\/pre><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/Winv.png\" alt=\"\"> <\/p><h4>Cholesky<a name=\"594e4474-66f7-4e9c-a5ee-4136c7f71922\"><\/a><\/h4><p>The matrix is symmetric.  Let's use <tt>chol<\/tt> to verify that it is positive definite.<\/p><pre class=\"codeinput\">    <span class=\"keyword\">try<\/span>\r\n        R = chol(W);\r\n    <span class=\"keyword\">catch<\/span>\r\n        disp(<span class=\"string\">\"Not SPD.\"<\/span>)\r\n    <span class=\"keyword\">end<\/span>\r\n<\/pre><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/R.png\" alt=\"\"> <\/p><p>Since <tt>chol<\/tt> was successful we know that <tt>W<\/tt> is positive definite. Factor out the diagonal to get a nice rational $LDL^T$ factorization.<\/p><pre class=\"codeinput\">    D = diag(diag(R));\r\n    L = R'\/D;\r\n    D = D^2;\r\n    LDLT = L*D*L';\r\n<\/pre><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/L.png\" alt=\"\"> <\/p><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/D.png\" alt=\"\"> <\/p><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/LDLT.png\" alt=\"\"> <\/p><h4>Eigenvalues<a name=\"ba455ab8-3cea-4ae9-8717-af0978569546\"><\/a><\/h4><p>The characteristic polynomial is a quartic with integer coefficients.<\/p><pre class=\"codeinput\">    p = charpoly(W,<span class=\"string\">'lambda'<\/span>);\r\n<\/pre><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/p.png\" alt=\"\"> <\/p><p>Since the degree of <i>p<\/i> is less than 5, the exact eigenvalues can be expressed in closed form.<\/p><pre class=\"codeinput\">    e = eig(W);\r\n<\/pre><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/eig.png\" alt=\"\"> <\/p><p>We asked for it.  Like it or not, that's the exact answer. Since <tt>p<\/tt> does not factor nicely over the rationals, the closed form is intimidating.  It's hard to tell which eigenvalue is the largest, for example.  Notice that even though the eigenvalues are real, the formulae involve complex numbers whose imaginary parts necessarily cancel.<\/p><p>For the remainder of this article, let's revert to floating point arithmetic.<\/p><pre class=\"codeinput\">    W = double(W)\r\n<\/pre><pre class=\"codeoutput\">W =\r\n     5     7     6     5\r\n     7    10     8     7\r\n     6     8    10     9\r\n     5     7     9    10\r\n<\/pre><p>Now the computed eigenvalues are approximate, but more useful.<\/p><pre class=\"codeinput\">    e = eig(W)\r\n<\/pre><pre class=\"codeoutput\">e =\r\n    0.0102\r\n    0.8431\r\n    3.8581\r\n   30.2887\r\n<\/pre><h4>Condition<a name=\"0c105ada-0a76-445e-b318-177105a15057\"><\/a><\/h4><p>Is this matrix poorly conditioned?<\/p><pre class=\"codeinput\">    condW = cond(W)\r\n<\/pre><pre class=\"codeoutput\">condW =\r\n   2.9841e+03\r\n<\/pre><p>So the condition is \"only\" about 3000.  That may not seem to be very badly conditioned.  But let's look more closely.  Set up a linear system where we know the exact solution.<\/p><pre class=\"codeinput\">   xact = ones(4,1);\r\n   b = W*xact\r\n<\/pre><pre class=\"codeoutput\">b =\r\n    23\r\n    32\r\n    33\r\n    31\r\n<\/pre><p>And solve it.<\/p><pre class=\"codeinput\">   x = W\\b\r\n<\/pre><pre class=\"codeoutput\">x =\r\n    1.0000\r\n    1.0000\r\n    1.0000\r\n    1.0000\r\n<\/pre><p>We expect to get all ones, and we do.  But now perturb the right-hand side a little.<\/p><pre class=\"codeinput\">   delta_b = [.01 -.01 .01 -.01]'\r\n   b_prime = b + delta_b\r\n<\/pre><pre class=\"codeoutput\">delta_b =\r\n    0.0100\r\n   -0.0100\r\n    0.0100\r\n   -0.0100\r\nb_prime =\r\n   23.0100\r\n   31.9900\r\n   33.0100\r\n   30.9900\r\n<\/pre><p>This perturbation changes the solution drastically.<\/p><pre class=\"codeinput\">   x_prime = W\\b_prime\r\n<\/pre><pre class=\"codeoutput\">x_prime =\r\n    1.8200\r\n    0.5000\r\n    0.8100\r\n    1.1100\r\n<\/pre><h4>An Experiment<a name=\"9461b772-cf36-47af-9f4c-cd4abb09d6b6\"><\/a><\/h4><p>I want to argue that Wilson's matrix <i>is<\/i> badly conditioned, when compared to matrices with these properties.<\/p><div><ul><li>4-by-4<\/li><li>Symmetric<\/li><li>Nonsingular<\/li><li>Integer entries between 1 and 10.<\/li><\/ul><\/div><p>My experiment involves generating one million random matrices with these properties.  Random symmetric matrices are created by<\/p><pre class=\"codeinput\">    U = triu(randi(10,4,4));\r\n    A = U + triu(U,1)';\r\n<\/pre><p>One typical run produced these statistics.<\/p><div><ul><li>Number of matrices = 1000000<\/li><li>Singular = 1820<\/li><li>Positive definite = 11217<\/li><li>Condition &gt; condW = 2115<\/li><li>Maximum condition = 4.8087e4<\/li><\/ul><\/div><p>The distribution of the condition numbers looks like this. We can see that Wilson's matrix is unusual. Only about 0.21 percent of comparable matrices have a larger condition number.<\/p><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/histo.png\" alt=\"\"> <\/p><p>By the way, the matrix with the largest condition number in this run was<\/p><pre class=\"codeinput\">    condMax\r\n<\/pre><pre class=\"codeoutput\">condMax =\r\n     1     3    10    10\r\n     3     4     8     9\r\n    10     8     3     9\r\n    10     9     9     3\r\n<\/pre><h4>Thanks<a name=\"8cf06111-dff0-4ec8-8abd-ef7364be477e\"><\/a><\/h4><p>Thanks to Tahar Loulou whose comment got this all started, to Steve Lord who contributed another comment, and to Jan Peter Sch&auml;fermeyer, Nick Higham and Sven Hammarling for their email pointing to references.<\/p><h4>References<a name=\"086a6209-ef3f-4153-8a00-8eb297f35f5c\"><\/a><\/h4><p>R. W. Zdanowich and T. S. Wilson, \"The elements of pendulum dampers,\" <i>Proceedings of the Institution of Mechanical Engineers<\/i>, 143_028_02, pp. 182-210, 1940, <a href=\"http:\/\/doi.org\/10.1243\/PIME_PROC_1940_143_028_02\">link<\/a>.<\/p><p>J. Morris, \"XVI. An escalator process for the solution of linear simultaneous equations\", <i>The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science<\/i>, 37:265, pp. 106-120, 1946, <a href=\"https:\/\/doi.org\/10.1080\/14786444608561331\">link<\/a>.<\/p><p>Jacques Delforge, \"Application of the concept of the conditioning of a matrix to an identification problem,\" <i>Comp. and Maths with Appls.<\/i>, Vol. 4, pp. 139-148, 1978, <a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/0898122178900238\">link<\/a>.<\/p><p>E. Bodewig, Matrix calculus, North Holland, Amsterdam, 2nd edition, 1959.<\/p><p>John Todd, \"The condition of a certain matrix,\" <i>Proceedings of the Cambridge Philosophical Society<\/i>, vol. 46, pp. 116-118, 1950, <a href=\"https:\/\/doi.org\/10.1017\/S0305004100025536\">link<\/a>.<\/p><p>John Todd, Basic Numerical Algorithms: Vol. 2: Numerical Algorithms, Birkh&auml;user Verlag, Basel, 1977.<\/p><p>Marvin Marcus, Basic Theorems in Matrix Theory, National Bureau of Standards, Applied Mathematics Series, number 57, page 22, 1960, <a href=\"https:\/\/babel.hathitrust.org\/cgi\/pt?id=mdp.39015023899019;view=1up;seq=3\">link<\/a>.<\/p><script language=\"JavaScript\"> <!-- \r\n    function grabCode_5d29a4477eec478d8c0df83e6297b284() {\r\n        \/\/ Remember the title so we can use it in the new page\r\n        title = document.title;\r\n\r\n        \/\/ Break up these strings so that their presence\r\n        \/\/ in the Javascript doesn't mess up the search for\r\n        \/\/ the MATLAB code.\r\n        t1='5d29a4477eec478d8c0df83e6297b284 ' + '##### ' + 'SOURCE BEGIN' + ' #####';\r\n        t2='##### ' + 'SOURCE END' + ' #####' + ' 5d29a4477eec478d8c0df83e6297b284';\r\n    \r\n        b=document.getElementsByTagName('body')[0];\r\n        i1=b.innerHTML.indexOf(t1)+t1.length;\r\n        i2=b.innerHTML.indexOf(t2);\r\n \r\n        code_string = b.innerHTML.substring(i1, i2);\r\n        code_string = code_string.replace(\/REPLACE_WITH_DASH_DASH\/g,'--');\r\n\r\n        \/\/ Use \/x3C\/g instead of the less-than character to avoid errors \r\n        \/\/ in the XML parser.\r\n        \/\/ Use '\\x26#60;' instead of '<' so that the XML parser\r\n        \/\/ doesn't go ahead and substitute the less-than character. \r\n        code_string = code_string.replace(\/\\x3C\/g, '\\x26#60;');\r\n\r\n        copyright = 'Copyright 2018 The MathWorks, Inc.';\r\n\r\n        w = window.open();\r\n        d = w.document;\r\n        d.write('<pre>\\n');\r\n        d.write(code_string);\r\n\r\n        \/\/ Add copyright line at the bottom if specified.\r\n        if (copyright.length > 0) {\r\n            d.writeln('');\r\n            d.writeln('%%');\r\n            if (copyright.length > 0) {\r\n                d.writeln('% _' + copyright + '_');\r\n            }\r\n        }\r\n\r\n        d.write('<\/pre>\\n');\r\n\r\n        d.title = title + ' (MATLAB code)';\r\n        d.close();\r\n    }   \r\n     --> <\/script><p style=\"text-align: right; font-size: xx-small; font-weight:lighter;   font-style: italic; color: gray\"><br><a href=\"javascript:grabCode_5d29a4477eec478d8c0df83e6297b284()\"><span style=\"font-size: x-small;        font-style: italic;\">Get \r\n      the MATLAB code <noscript>(requires JavaScript)<\/noscript><\/span><\/a><br><br>\r\n      Published with MATLAB&reg; R2018a<br><\/p><\/div><!--\r\n5d29a4477eec478d8c0df83e6297b284 ##### SOURCE BEGIN #####\r\n%% Reviving T. S. Wilson's Matrix\r\n% I probably first saw this matrix in 1960 in John Todd's class at Caltech.\r\n% But I forgot about it until Tahar Loulou jogged my memory with a \r\n% comment following\r\n% <https:\/\/blogs.mathworks.com\/cleve\/2018\/05\/28\/the-jordan-canonical-form-just-doesnt-compute \r\n% my blog post in late May>.\r\n% It deserves a place in our gallery of interesting matrices.\r\n\r\n%% T. S. Wilson\r\n% I'm afraid that I don't know very much about T. S. Wilson himself.\r\n% The only paper we could find on the Internet is the joint paper\r\n% about pendulum dampers listed in the references.  This paper, which\r\n% does not mention the matrix, was published in 1940.  The authors\r\n% were from a British company called D. Napier & Son Ltd.\r\n% The earliest appearance of the matrix that we know of  is the 1946 \r\n% paper by J. Morris referenced below.  It is possible that by this\r\n% time Wilson and Morris were colleagues at the Royal Aircraft \r\n% Establishment in Farnborough in the UK.\r\n%\r\n% I think that Wilson and Todd must have known each other.\r\n% Todd acknowledges Wilson in several papers and books.\r\n%\r\n% One of the references we used in Todd's class is the booklet written\r\n% by Marvin Marcus also referenced below.  The booklet has a short\r\n% section collecting some \"particular\" matrices.  A_13 is our old friend\r\n% the Hilbert matrix and A_14 is Wilson's matrix.\r\n\r\n%% T. S. Wilson's Matrix\r\n% Here is the matrix.\r\n% Let's use the Live Editor(R) and the Symbolic Math Toolbox(R).\r\n\r\n    W = sym([5 7 6 5; 7 10 8 7; 6 8 10 9; 5 7 9 10]);\r\n    \r\n%%\r\n% <<W.png>>\r\n\r\n%% Determinant and Inverse\r\n% The first noteworthy property is the determinant.  The determinant has\r\n% to be an integer because all the matrix elements are integers.\r\n \r\n    d = det(W);\r\n    \r\n%%\r\n% <<det.png>>\r\n\r\n%%\r\n% This is the smallest possible determinant for a nonsingular matrix\r\n% with integer entries.\r\n\r\n%%\r\n% Since the determinant is one, the inverse must also have integer \r\n% entries.\r\n\r\n    Winv = round(inv(W));\r\n    \r\n%%\r\n% <<Winv.png>>\r\n    \r\n%% Cholesky\r\n% The matrix is symmetric.  Let's use |chol| to verify that it is\r\n% positive definite.\r\n\r\n    try\r\n        R = chol(W);\r\n    catch\r\n        disp(\"Not SPD.\")\r\n    end\r\n    \r\n%%\r\n% <<R.png>>\r\n    \r\n%%\r\n% Since |chol| was successful we know that |W| is positive definite.\r\n% Factor out the diagonal to get a nice rational $LDL^T$ factorization.\r\n\r\n    D = diag(diag(R));\r\n    L = R'\/D; \r\n    D = D^2;\r\n    LDLT = L*D*L';\r\n    \r\n%%\r\n% <<L.png>>\r\n%\r\n% <<D.png>>\r\n%\r\n% <<LDLT.png>>\r\n    \r\n%% Eigenvalues\r\n% The characteristic polynomial is a quartic with integer\r\n% coefficients.\r\n\r\n    p = charpoly(W,'lambda');\r\n    \r\n%%\r\n% <<p.png>>\r\n\r\n%%\r\n% Since the degree of _p_ is less than 5,\r\n% the exact eigenvalues can be expressed in closed form.\r\n\r\n    e = eig(W);\r\n    \r\n%%\r\n% <<eig.png>>\r\n    \r\n%%\r\n% We asked for it.  Like it or not, that's the exact answer.\r\n% Since |p| does not factor nicely over the rationals, the closed\r\n% form is intimidating.  It's hard to tell which eigenvalue is the\r\n% largest, for example.  Notice that even though the eigenvalues are real,\r\n% the formulae involve complex numbers whose imaginary parts\r\n% necessarily cancel.\r\n\r\n%%\r\n% For the remainder of this article, let's revert to floating point\r\n% arithmetic.\r\n\r\n    W = double(W)\r\n    \r\n%%\r\n% Now the computed eigenvalues are approximate, but more useful.\r\n\r\n    e = eig(W)\r\n\r\n%% Condition\r\n% Is this matrix poorly conditioned?\r\n\r\n    condW = cond(W)\r\n    \r\n%%\r\n% So the condition is \"only\" about 3000.  That may not seem to be very\r\n% badly conditioned.  But let's look more closely.  Set up a linear system\r\n% where we know the exact solution.\r\n\r\n   xact = ones(4,1);\r\n   b = W*xact\r\n   \r\n%%\r\n% And solve it.\r\n\r\n   x = W\\b\r\n   \r\n%%\r\n% We expect to get all ones, and we do.  But now perturb the right-hand\r\n% side a little.\r\n\r\n   delta_b = [.01 -.01 .01 -.01]'\r\n   b_prime = b + delta_b\r\n   \r\n%%\r\n% This perturbation changes the solution drastically.\r\n\r\n   x_prime = W\\b_prime\r\n    \r\n%% An Experiment\r\n% I want to argue that Wilson's matrix _is_ badly conditioned,\r\n% when compared to matrices with these properties.\r\n%\r\n% * 4-by-4\r\n% * Symmetric\r\n% * Nonsingular\r\n% * Integer entries between 1 and 10.\r\n\r\n%%\r\n% My experiment involves generating one million random matrices\r\n% with these properties.  Random symmetric matrices are created by\r\n\r\n    U = triu(randi(10,4,4));\r\n    A = U + triu(U,1)';\r\n    \r\n%%\r\n% One typical run produced these statistics.\r\n%\r\n% * Number of matrices = 1000000\r\n% * Singular = 1820\r\n% * Positive definite = 11217\r\n% * Condition > condW = 2115\r\n% * Maximum condition = 4.8087e4\r\n\r\n%%\r\n% The distribution of the condition numbers looks like this.\r\n% We can see that Wilson's matrix is unusual.\r\n% Only about 0.21 percent of comparable matrices\r\n% have a larger condition number.\r\n%\r\n% <<histo.png>>\r\n\r\n%%\r\n% By the way, the matrix with the largest condition number in this run was\r\n\r\n    condMax\r\n\r\n%% Thanks\r\n% Thanks to Tahar Loulou whose comment got this all started, \r\n% to Steve Lord who contributed another comment,\r\n% and to Jan Peter Sch\u00c3\u00a4fermeyer, Nick Higham and Sven Hammarling \r\n% for their email pointing to references.\r\n\r\n%% References\r\n% R. W. Zdanowich and T. S. Wilson, \"The elements of pendulum dampers,\"\r\n% _Proceedings of the Institution of Mechanical Engineers_, 143_028_02,\r\n% pp. 182-210, 1940,\r\n% <http:\/\/doi.org\/10.1243\/PIME_PROC_1940_143_028_02 link>.\r\n%\r\n% J. Morris, \"XVI. An escalator process for the solution of linear\r\n% simultaneous equations\", _The London, Edinburgh, and Dublin\r\n% Philosophical Magazine and Journal of Science_, 37:265, pp. 106-120,\r\n% 1946, <https:\/\/doi.org\/10.1080\/14786444608561331 link>.\r\n%\r\n% Jacques Delforge, \"Application of the concept of the conditioning\r\n% of a matrix to an identification problem,\" _Comp. and Maths with Appls._,\r\n% Vol. 4, pp. 139-148, 1978,\r\n% <https:\/\/www.sciencedirect.com\/science\/article\/pii\/0898122178900238\r\n% link>.\r\n%\r\n% E. Bodewig, Matrix calculus, North Holland, Amsterdam,\r\n% 2nd edition, 1959.\r\n%\r\n% John Todd, \"The condition of a certain matrix,\"\r\n% _Proceedings of the Cambridge Philosophical Society_, vol. 46,\r\n% pp. 116-118, 1950, <https:\/\/doi.org\/10.1017\/S0305004100025536 link>.\r\n%\r\n% John Todd, Basic Numerical Algorithms: Vol. 2: Numerical Algorithms,\r\n% Birkh\u00c3\u00a4user Verlag, Basel, 1977.\r\n% \r\n% Marvin Marcus, Basic Theorems in Matrix Theory, National Bureau of\r\n% Standards, Applied Mathematics Series, number 57, page 22, 1960,\r\n% <https:\/\/babel.hathitrust.org\/cgi\/pt?id=mdp.39015023899019;view=1up;seq=3\r\n% link>.\r\n##### SOURCE END ##### 5d29a4477eec478d8c0df83e6297b284\r\n-->","protected":false},"excerpt":{"rendered":"<div class=\"overview-image\"><img src=\"https:\/\/blogs.mathworks.com\/cleve\/files\/Wbox.png\" class=\"img-responsive attachment-post-thumbnail size-post-thumbnail wp-post-image\" alt=\"\" decoding=\"async\" loading=\"lazy\" \/><\/div><!--introduction--><p>I probably first saw this matrix in 1960 in John Todd's class at Caltech. But I forgot about it until Tahar Loulou jogged my memory with a comment following <a href=\"https:\/\/blogs.mathworks.com\/cleve\/2018\/05\/28\/the-jordan-canonical-form-just-doesnt-compute\">my blog post in late May<\/a>. It deserves a place in our gallery of interesting matrices.... <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/cleve\/2018\/08\/20\/reviving-wilsons-matrix\/\">read more >><\/a><\/p>","protected":false},"author":78,"featured_media":3755,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[4,6,16,8],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/3701"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/users\/78"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/comments?post=3701"}],"version-history":[{"count":4,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/3701\/revisions"}],"predecessor-version":[{"id":3753,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/3701\/revisions\/3753"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/media\/3755"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/media?parent=3701"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/categories?post=3701"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/tags?post=3701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}