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New analytic and combinatorial tools make it possible to solve the design problem.
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内容介绍
<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/624f79936caf596192dd4e65/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/624f79936caf596192dd4e65/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">CS229<span class="_ _0"> </span>Problem<span class="_ _0"> </span>Set<span class="_ _0"> </span>#0<span class="_ _1"> </span><span class="ff2">1</span></div><div class="t m0 x1 h3 y2 ff3 fs1 fc0 sc0 ls0 ws0">CS<span class="_ _2"> </span>229,<span class="_ _2"> </span>F<span class="_ _3"></span>all<span class="_ _2"> </span>2018</div><div class="t m0 x1 h3 y3 ff3 fs1 fc0 sc0 ls0 ws0">Problem<span class="_ _4"> </span>Set<span class="_ _4"> </span>#0:<span class="_ _5"> </span>Linear<span class="_ _4"> </span>Algebra<span class="_ _4"> </span>and<span class="_ _4"> </span>Multiv<span class="_ _6"></span>ariable</div><div class="t m0 x1 h3 y4 ff3 fs1 fc0 sc0 ls0 ws0">Calculus</div><div class="t m0 x1 h2 y5 ff4 fs0 fc0 sc0 ls0 ws0">Notes:<span class="_ _7"> </span><span class="ff2">(1)<span class="_ _8"> </span>These<span class="_ _8"> </span>questions<span class="_ _8"> </span>require<span class="_ _8"> </span>though<span class="_ _9"></span>t,<span class="_ _4"> </span>but<span class="_ _8"> </span>do<span class="_ _8"> </span>not<span class="_ _8"> </span>require<span class="_ _8"> </span>long<span class="_ _8"> </span>answ<span class="_ _9"></span>ers.<span class="_ _5"> </span>Please<span class="_ _8"> </span>b<span class="_ _a"></span>e<span class="_ _8"> </span>as</span></div><div class="t m0 x1 h2 y6 ff2 fs0 fc0 sc0 ls0 ws0">concise<span class="_ _0"> </span>as<span class="_ _b"> </span>p<span class="_ _a"></span>ossible.<span class="_ _c"> </span>(2)<span class="_ _0"> </span>If<span class="_ _b"> </span>y<span class="_ _9"></span>ou<span class="_ _0"> </span>hav<span class="_ _9"></span>e<span class="_ _0"> </span>a<span class="_ _b"> </span>question<span class="_ _0"> </span>ab<span class="_ _a"></span>out<span class="_ _0"> </span>this<span class="_ _b"> </span>homework,<span class="_ _0"> </span>w<span class="_ _9"></span>e<span class="_ _0"> </span>encourage<span class="_ _b"> </span>y<span class="_ _9"></span>ou<span class="_ _0"> </span>to<span class="_ _b"> </span>p<span class="_ _a"></span>ost</div><div class="t m0 x1 h2 y7 ff2 fs0 fc0 sc0 ls0 ws0">y<span class="_ _9"></span>our<span class="_"> </span>question<span class="_"> </span>on<span class="_ _0"> </span>our<span class="_"> </span>Piazza<span class="_"> </span>forum,<span class="_ _d"> </span>at<span class="_"> </span><span class="ff5">https://piazza.com/stanford/fall2018/cs229</span>.<span class="_ _8"> </span>(3)<span class="_"> </span>If</div><div class="t m0 x1 h2 y8 ff2 fs0 fc0 sc0 ls0 ws0">y<span class="_ _9"></span>ou<span class="_ _d"> </span>missed<span class="_ _d"> </span>the<span class="_ _d"> </span>&#64257;rst<span class="_ _0"> </span>lecture<span class="_ _d"> </span>or<span class="_ _d"> </span>are<span class="_ _d"> </span>unfamiliar<span class="_ _0"> </span>with<span class="_ _d"> </span>the<span class="_ _d"> </span>collab<span class="_ _a"></span>oration<span class="_ _d"> </span>or<span class="_ _d"> </span>honor<span class="_ _0"> </span>co<span class="_ _a"></span>de<span class="_ _d"> </span>p<span class="_ _a"></span>olicy<span class="_ _e"></span>,<span class="_ _d"> </span>please</div><div class="t m0 x1 h2 y9 ff2 fs0 fc0 sc0 ls0 ws0">read<span class="_ _f"> </span>the<span class="_ _f"> </span>p<span class="_ _a"></span>olicy<span class="_ _f"> </span>on<span class="_ _f"> </span>Handout<span class="_ _f"> </span>#1<span class="_ _f"> </span>(a<span class="_ _9"></span>v<span class="_ _e"></span>ailable<span class="_ _f"> </span>from<span class="_ _f"> </span>the<span class="_ _f"> </span>course<span class="_ _f"> </span>website)<span class="_ _f"> </span>before<span class="_ _f"> </span>starting<span class="_ _f"> </span>work.<span class="_ _7"> </span>(4)</div><div class="t m0 x1 h2 ya ff2 fs0 fc0 sc0 ls0 ws0">This<span class="_ _f"> </span>sp<span class="_ _a"></span>eci&#64257;c<span class="_ _f"> </span>homew<span class="_ _9"></span>ork<span class="_ _f"> </span>is<span class="_ _f"> </span><span class="ff6">not<span class="_ _f"> </span>gr<span class="_ _9"></span>ade<span class="_ _e"></span>d<span class="ff2">,<span class="_ _8"> </span>but<span class="_ _f"> </span>w<span class="_ _9"></span>e<span class="_ _f"> </span>encourage<span class="_ _f"> </span>y<span class="_ _9"></span>ou<span class="_ _f"> </span>to<span class="_ _f"> </span>solv<span class="_ _9"></span>e<span class="_ _f"> </span>eac<span class="_ _9"></span>h<span class="_ _b"> </span>of<span class="_ _f"> </span>the<span class="_ _f"> </span>problems<span class="_ _f"> </span>to</span></span></div><div class="t m0 x1 h2 yb ff2 fs0 fc0 sc0 ls0 ws0">brush<span class="_ _d"> </span>up<span class="_ _d"> </span>on<span class="_ _d"> </span>your<span class="_ _d"> </span>linear<span class="_ _d"> </span>algebra.<span class="_ _8"> </span>Some<span class="_ _d"> </span>of<span class="_ _d"> </span>them<span class="_ _0"> </span>ma<span class="_ _9"></span>y<span class="_ _d"> </span>even<span class="_"> </span>b<span class="_ _a"></span>e<span class="_ _d"> </span>useful<span class="_ _0"> </span>for<span class="_ _d"> </span>subsequent<span class="_"> </span>problem<span class="_ _d"> </span>sets.</div><div class="t m0 x1 h2 yc ff2 fs0 fc0 sc0 ls0 ws0">It<span class="_ _0"> </span>also<span class="_ _0"> </span>serv<span class="_ _9"></span>es<span class="_ _0"> </span>as<span class="_ _0"> </span>y<span class="_ _9"></span>our<span class="_ _0"> </span>in<span class="_ _9"></span>tro<span class="_ _a"></span>duction<span class="_ _d"> </span>to<span class="_ _0"> </span>using<span class="_ _0"> </span>Gradescop<span class="_ _a"></span>e<span class="_ _0"> </span>for<span class="_ _0"> </span>submissions.</div><div class="t m0 x1 h2 yd ff2 fs0 fc0 sc0 ls0 ws0">1.<span class="_ _c"> </span><span class="ff4">[0<span class="_ _b"> </span>p<span class="_ _a"></span>oints]<span class="_"> </span>Gradien<span class="_ _9"></span>ts<span class="_ _b"> </span>and<span class="_ _b"> </span>Hessians</span></div><div class="t m0 x2 h2 ye ff2 fs0 fc0 sc0 ls0 ws0">Recall<span class="_ _b"> </span>that<span class="_ _0"> </span>a<span class="_ _b"> </span>matrix<span class="_ _b"> </span><span class="ff7">A<span class="_ _d"> </span><span class="ff8">&#8712;<span class="_ _0"> </span><span class="ff9">R</span></span></span></div><div class="t m0 x3 h4 yf ffa fs2 fc0 sc0 ls0 ws0">n<span class="ffb">&#215;</span>n</div><div class="t m0 x4 h2 ye ff2 fs0 fc0 sc0 ls0 ws0">is<span class="_ _b"> </span><span class="ff6">symmetric<span class="_ _f"> </span></span>if<span class="_ _0"> </span><span class="ff7">A</span></div><div class="t m0 x5 h4 yf ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x6 h2 ye ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _d"> </span><span class="ff7">A</span>,<span class="_ _b"> </span>that<span class="_ _b"> </span>is,<span class="_ _b"> </span><span class="ff7">A</span></div><div class="t m0 x7 h4 y10 ffa fs2 fc0 sc0 ls0 ws0">ij</div><div class="t m0 x8 h2 ye ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _d"> </span><span class="ff7">A</span></div><div class="t m0 x9 h4 y10 ffa fs2 fc0 sc0 ls0 ws0">j<span class="_ _a"></span>i</div><div class="t m0 xa h2 ye ff2 fs0 fc0 sc0 ls0 ws0">for<span class="_ _b"> </span>all<span class="_ _0"> </span><span class="ff7">i,<span class="_ _10"> </span>j<span class="_ _a"></span></span>.<span class="_ _11"> </span>Also</div><div class="t m0 x2 h2 y11 ff2 fs0 fc0 sc0 ls0 ws0">recall<span class="_"> </span>the<span class="_ _12"> </span>gradien<span class="_ _9"></span>t<span class="_"> </span><span class="ff8">&#8711;<span class="ff7">f<span class="_ _13"></span></span></span>(<span class="ff7">x</span>)<span class="_"> </span>of<span class="_ _12"> </span>a<span class="_"> </span>function<span class="_ _12"> </span><span class="ff7">f<span class="_ _b"> </span></span>:<span class="_"> </span><span class="ff9">R</span></div><div class="t m0 xb h4 y12 ffa fs2 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc h2 y11 ff8 fs0 fc0 sc0 ls0 ws0">&#8594;<span class="_ _14"> </span><span class="ff9">R<span class="ff2">,<span class="_"> </span>whic<span class="_ _9"></span>h<span class="_"> </span>is<span class="_ _12"> </span>the<span class="_"> </span><span class="ff7">n</span>-v<span class="_ _9"></span>ector<span class="_ _12"> </span>of<span class="_"> </span>partial<span class="_ _12"> </span>deriv<span class="_ _e"></span>atives</span></span></div><div class="t m0 xd h2 y13 ff8 fs0 fc0 sc0 ls0 ws0">&#8711;<span class="ff7">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)<span class="_"> </span>=</span></span></div><div class="t m0 xe h5 y14 ffc fs0 fc0 sc0 ls0 ws0">&#63726;</div><div class="t m0 xe h5 y15 ffc fs0 fc0 sc0 ls0 ws0">&#63727;</div><div class="t m0 xe h5 y16 ffc fs0 fc0 sc0 ls0 ws0">&#63728;</div><div class="t m0 xf h4 y17 ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x10 h4 y18 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x11 h6 y19 ffd fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x12 h2 y1a ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x12 h2 y1b ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x12 h2 y1c ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x12 h2 y1d ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 xf h4 y1e ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x10 h4 y1f ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x11 h7 y20 ffe fs3 fc0 sc0 ls0 ws0">n</div><div class="t m0 x12 h2 y21 ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x13 h5 y14 ffc fs0 fc0 sc0 ls0 ws0">&#63737;</div><div class="t m0 x13 h5 y15 ffc fs0 fc0 sc0 ls0 ws0">&#63738;</div><div class="t m0 x13 h5 y16 ffc fs0 fc0 sc0 ls0 ws0">&#63739;</div><div class="t m0 x14 h2 y13 ff2 fs0 fc0 sc0 ls0 ws0">where<span class="_ _7"> </span><span class="ff7">x<span class="_ _14"> </span></span>=</div><div class="t m0 x15 h5 y14 ffc fs0 fc0 sc0 ls0 ws0">&#63726;</div><div class="t m0 x15 h5 y15 ffc fs0 fc0 sc0 ls0 ws0">&#63727;</div><div class="t m0 x15 h5 y16 ffc fs0 fc0 sc0 ls0 ws0">&#63728;</div><div class="t m0 x16 h8 y22 ff7 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x17 h9 y23 fff fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x18 h2 y1b ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x18 h2 y1c ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x18 h2 y1d ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x16 h8 y24 ff7 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x17 h4 y25 ffa fs2 fc0 sc0 ls0 ws0">n</div><div class="t m0 x19 h5 y14 ffc fs0 fc0 sc0 ls0 ws0">&#63737;</div><div class="t m0 x19 h5 y15 ffc fs0 fc0 sc0 ls0 ws0">&#63738;</div><div class="t m0 x19 h5 y16 ffc fs0 fc0 sc0 ls0 ws0">&#63739;</div><div class="t m0 x1a h8 y13 ff7 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x2 h2 y26 ff2 fs0 fc0 sc0 ls0 ws0">The<span class="_ _0"> </span>hessian<span class="_ _d"> </span><span class="ff8">&#8711;</span></div><div class="t m0 x1b h9 y27 fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1c h2 y26 ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)<span class="_ _0"> </span>of<span class="_ _0"> </span>a<span class="_ _0"> </span>function<span class="_ _0"> </span></span>f<span class="_ _b"> </span><span class="ff2">:<span class="_"> </span><span class="ff9">R</span></span></div><div class="t m0 x11 h4 y27 ffa fs2 fc0 sc0 ls0 ws0">n</div><div class="t m0 x1d h2 y26 ff8 fs0 fc0 sc0 ls0 ws0">&#8594;<span class="_ _14"> </span><span class="ff9">R<span class="_ _0"> </span><span class="ff2">is<span class="_ _d"> </span>the<span class="_ _0"> </span><span class="ff7">n<span class="_ _12"> </span></span></span></span>&#215;<span class="_ _15"> </span><span class="ff7">n<span class="_ _0"> </span><span class="ff2">symmetric<span class="_ _d"> </span>matrix<span class="_ _0"> </span>of<span class="_ _0"> </span>twice<span class="_ _d"> </span>partial</span></span></div><div class="t m0 x2 h2 y28 ff2 fs0 fc0 sc0 ls0 ws0">deriv<span class="_ _16"></span>ativ<span class="_ _9"></span>es,</div><div class="t m0 x1e h2 y29 ff8 fs0 fc0 sc0 ls0 ws0">&#8711;</div><div class="t m0 x1f h9 y2a fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x20 h2 y29 ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)<span class="_"> </span>=</span></div><div class="t m0 x21 h5 y2b ffc fs0 fc0 sc0 ls0 ws0">&#63726;</div><div class="t m0 x21 h5 y2c ffc fs0 fc0 sc0 ls0 ws0">&#63727;</div><div class="t m0 x21 h5 y2d ffc fs0 fc0 sc0 ls0 ws0">&#63727;</div><div class="t m0 x21 h5 y2e ffc fs0 fc0 sc0 ls0 ws0">&#63727;</div><div class="t m0 x21 h5 y2f ffc fs0 fc0 sc0 ls0 ws0">&#63727;</div><div class="t m0 x21 h5 y30 ffc fs0 fc0 sc0 ls0 ws0">&#63727;</div><div class="t m0 x21 h5 y31 ffc fs0 fc0 sc0 ls0 ws0">&#63728;</div><div class="t m0 x22 h4 y32 ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x23 h6 y33 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x24 h4 y34 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x25 h6 y35 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x25 h6 y36 ffd fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x26 h2 y37 ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x13 h4 y32 ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x27 h6 y33 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x28 h4 y38 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 xc h6 y39 ffd fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x27 h4 y38 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x29 h6 y39 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2a h2 y37 ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)<span class="_ _17"> </span><span class="ff8">&#183;<span class="_ _10"> </span>&#183;<span class="_ _18"> </span>&#183;</span></span></div><div class="t m0 x2b h4 y32 ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x2c h6 y33 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x16 h4 y38 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x2b h6 y39 ffd fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x2d h4 y38 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x2e h7 y39 ffe fs3 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2f h2 y37 ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x22 h4 y3a ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x23 h6 y3b ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x30 h4 y3c ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x22 h6 y3d ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x4 h4 y3c ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x31 h6 y3d ffd fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x32 h2 y3e ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x13 h4 y3a ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x27 h6 y3b ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x33 h4 y3f ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x34 h6 y40 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x34 h6 y41 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x35 h2 y3e ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)<span class="_ _19"> </span><span class="ff8">&#183;<span class="_ _18"> </span>&#183;<span class="_ _10"> </span>&#183;</span></span></div><div class="t m0 x2b h4 y3a ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x2c h6 y3b ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x16 h4 y3c ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x2b h6 y3d ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2d h4 y3c ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x2e h7 y3d ffe fs3 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2f h2 y3e ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x26 h2 y42 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x26 h2 y43 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x26 h2 y44 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x35 h2 y42 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x35 h2 y43 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x35 h2 y44 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x36 h2 y45 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x37 h2 y43 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x38 h2 y46 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x2e h2 y42 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x2e h2 y43 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x2e h2 y44 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x22 h4 y47 ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x23 h6 y48 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x30 h4 y49 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x22 h7 y4a ffe fs3 fc0 sc0 ls0 ws0">n</div><div class="t m0 x23 h4 y49 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x31 h6 y4a ffd fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 x32 h2 y4b ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x13 h4 y47 ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x27 h6 y48 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x28 h4 y49 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x13 h7 y4a ffe fs3 fc0 sc0 ls0 ws0">n</div><div class="t m0 x27 h4 y49 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x29 h6 y4a ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2a h2 y4b ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)<span class="_ _1a"> </span><span class="ff8">&#183;<span class="_ _18"> </span>&#183;<span class="_ _10"> </span>&#183;</span></span></div><div class="t m0 x2b h4 y47 ffa fs2 fc0 sc0 ls0 ws0">&#8706;</div><div class="t m0 x2c h6 y48 ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x39 h4 y49 ffa fs2 fc0 sc0 ls0 ws0">&#8706;<span class="_ _a"></span>x</div><div class="t m0 x3a h6 y4c ffd fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3a h7 y4d ffe fs3 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2e h2 y4b ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x3b h5 y2b ffc fs0 fc0 sc0 ls0 ws0">&#63737;</div><div class="t m0 x3b h5 y2c ffc fs0 fc0 sc0 ls0 ws0">&#63738;</div><div class="t m0 x3b h5 y2d ffc fs0 fc0 sc0 ls0 ws0">&#63738;</div><div class="t m0 x3b h5 y2e ffc fs0 fc0 sc0 ls0 ws0">&#63738;</div><div class="t m0 x3b h5 y2f ffc fs0 fc0 sc0 ls0 ws0">&#63738;</div><div class="t m0 x3b h5 y30 ffc fs0 fc0 sc0 ls0 ws0">&#63738;</div><div class="t m0 x3b h5 y31 ffc fs0 fc0 sc0 ls0 ws0">&#63739;</div><div class="t m0 x3c h8 y29 ff7 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x2 h2 y4e ff2 fs0 fc0 sc0 ls0 ws0">(a)<span class="_ _c"> </span>Let<span class="_ _0"> </span><span class="ff7">f<span class="_ _1b"> </span></span>(<span class="ff7">x</span>)<span class="_"> </span>=</div><div class="t m0 x3d h9 y4f fff fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x3d h9 y50 fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1e h8 y4e ff7 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x3e h4 y51 ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x3f h2 y4e ff7 fs0 fc0 sc0 ls0 ws0">Ax<span class="_ _15"> </span><span class="ff2">+<span class="_ _12"> </span></span>b</div><div class="t m0 x40 h4 y51 ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x41 h2 y4e ff7 fs0 fc0 sc0 ls0 ws0">x<span class="ff2">,<span class="_ _0"> </span>where<span class="_ _0"> </span></span>A<span class="_ _0"> </span><span class="ff2">is<span class="_ _b"> </span>a<span class="_ _d"> </span>symmetric<span class="_ _b"> </span>matrix<span class="_ _d"> </span>and<span class="_ _b"> </span></span>b<span class="_ _14"> </span><span class="ff8">&#8712;<span class="_ _14"> </span><span class="ff9">R</span></span></div><div class="t m0 x3b h4 y51 ffa fs2 fc0 sc0 ls0 ws0">n</div><div class="t m0 x3c h2 y4e ff2 fs0 fc0 sc0 ls0 ws0">is<span class="_ _0"> </span>a<span class="_ _0"> </span>vector.<span class="_ _8"> </span>What</div><div class="t m0 x42 h2 y52 ff2 fs0 fc0 sc0 ls0 ws0">is<span class="_ _0"> </span><span class="ff8">&#8711;<span class="ff7">f<span class="_ _13"></span></span></span>(<span class="ff7">x</span>)?</div><div class="t m0 x2 h2 y53 ff2 fs0 fc0 sc0 ls0 ws0">(b)<span class="_ _c"> </span>Let<span class="_ _b"> </span><span class="ff7">f<span class="_ _13"></span></span>(<span class="ff7">x</span>)<span class="_ _b"> </span>=<span class="_ _0"> </span><span class="ff7">g<span class="_ _a"></span></span>(<span class="ff7">h</span>(<span class="ff7">x</span>)),<span class="_ _b"> </span>where<span class="_ _b"> </span><span class="ff7">g<span class="_ _b"> </span></span>:<span class="_ _b"> </span><span class="ff9">R<span class="_ _0"> </span><span class="ff8">&#8594;<span class="_ _0"> </span></span>R<span class="_ _b"> </span></span>is<span class="_ _b"> </span>di&#64256;erentiable<span class="_ _0"> </span>and<span class="_ _b"> </span><span class="ff7">h<span class="_ _b"> </span></span>:<span class="_ _0"> </span><span class="ff9">R</span></div><div class="t m0 x43 h4 y54 ffa fs2 fc0 sc0 ls0 ws0">n</div><div class="t m0 x44 h2 y53 ff8 fs0 fc0 sc0 ls0 ws0">&#8594;<span class="_ _0"> </span><span class="ff9">R<span class="_ _b"> </span><span class="ff2">is<span class="_ _b"> </span>di&#64256;erentiable.</span></span></div><div class="t m0 x42 h2 y55 ff2 fs0 fc0 sc0 ls0 ws0">What<span class="_ _0"> </span>is<span class="_ _0"> </span><span class="ff8">&#8711;<span class="ff7">f<span class="_ _13"></span></span></span>(<span class="ff7">x</span>)?</div><div class="t m0 x2 h2 y56 ff2 fs0 fc0 sc0 ls0 ws0">(c)<span class="_ _c"> </span>Let<span class="_"> </span><span class="ff7">f<span class="_ _13"></span></span>(<span class="ff7">x</span>)<span class="_"> </span>=</div><div class="t m0 x45 h9 y57 fff fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x45 h9 y58 fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x46 h8 y56 ff7 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x47 h4 y59 ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x48 h2 y56 ff7 fs0 fc0 sc0 ls0 ws0">Ax<span class="_ _1b"> </span><span class="ff2">+<span class="_ _1b"> </span></span>b</div><div class="t m0 x49 h4 y59 ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x21 h2 y56 ff7 fs0 fc0 sc0 ls0 ws0">x<span class="ff2">,<span class="_"> </span>where<span class="_"> </span></span>A<span class="_ _d"> </span><span class="ff2">is<span class="_"> </span>symmetric<span class="_"> </span>and<span class="_"> </span></span>b<span class="_ _14"> </span><span class="ff8">&#8712;<span class="_ _14"> </span><span class="ff9">R</span></span></div><div class="t m0 x4a h4 y59 ffa fs2 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2b h2 y56 ff2 fs0 fc0 sc0 ls0 ws0">is<span class="_"> </span>a<span class="_"> </span>vector.<span class="_ _f"> </span>What<span class="_"> </span>is<span class="_"> </span><span class="ff8">&#8711;</span></div><div class="t m0 x4b h9 y59 fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x4c h2 y56 ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)?</span></div><div class="t m0 x2 h2 y5a ff2 fs0 fc0 sc0 ls0 ws0">(d)<span class="_ _c"> </span>Let<span class="_"> </span><span class="ff7">f<span class="_ _1b"> </span></span>(<span class="ff7">x</span>)<span class="_"> </span>=<span class="_"> </span><span class="ff7">g<span class="_ _a"></span></span>(<span class="ff7">a</span></div><div class="t m0 x4d h4 y5b ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x4e h2 y5a ff7 fs0 fc0 sc0 ls0 ws0">x<span class="ff2">),<span class="_"> </span>where<span class="_ _d"> </span></span>g<span class="_ _d"> </span><span class="ff2">:<span class="_"> </span><span class="ff9">R<span class="_ _14"> </span><span class="ff8">&#8594;<span class="_ _14"> </span></span>R<span class="_ _14"> </span></span>is<span class="_ _d"> </span>con<span class="_ _9"></span>tin<span class="_ _9"></span>uously<span class="_"> </span>di&#64256;erentiable<span class="_"> </span>and<span class="_"> </span><span class="ff7">a<span class="_ _14"> </span><span class="ff8">&#8712;<span class="_ _14"> </span><span class="ff9">R</span></span></span></span></div><div class="t m0 x4f h4 y5b ffa fs2 fc0 sc0 ls0 ws0">n</div><div class="t m0 x50 h2 y5a ff2 fs0 fc0 sc0 ls0 ws0">is<span class="_"> </span>a<span class="_"> </span>vector.</div><div class="t m0 x42 h2 y5c ff2 fs0 fc0 sc0 ls0 ws0">What<span class="_ _d"> </span>are<span class="_ _0"> </span><span class="ff8">&#8711;<span class="ff7">f<span class="_ _1b"> </span></span></span>(<span class="ff7">x</span>)<span class="_ _0"> </span>and<span class="_ _0"> </span><span class="ff8">&#8711;</span></div><div class="t m0 x21 h9 y5d fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x51 h2 y5c ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)?<span class="_ _8"> </span>(<span class="ff6">Hint:<span class="_ _c"> </span></span>your<span class="_ _d"> </span>expression<span class="_ _0"> </span>for<span class="_ _0"> </span><span class="ff8">&#8711;</span></span></div><div class="t m0 x2b h9 y5d fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2c h2 y5c ff7 fs0 fc0 sc0 ls0 ws0">f<span class="_ _13"></span><span class="ff2">(</span>x<span class="ff2">)<span class="_ _0"> </span>may<span class="_ _d"> </span>hav<span class="_ _16"></span>e<span class="_ _0"> </span>as<span class="_ _0"> </span>few<span class="_ _d"> </span>as<span class="_ _0"> </span>11</span></div><div class="t m0 x42 h2 y5e ff2 fs0 fc0 sc0 ls0 ws0">sym<span class="_ _9"></span>b<span class="_ _a"></span>ols,<span class="_ _d"> </span>including</div><div class="t m0 x52 ha y5f ffb fs2 fc0 sc0 ls0 ws0">0</div><div class="t m0 x53 h2 y5e ff2 fs0 fc0 sc0 ls0 ws0">and<span class="_ _0"> </span>paren<span class="_ _9"></span>theses.)</div><div class="t m0 x1 h2 y60 ff2 fs0 fc0 sc0 ls0 ws0">2.<span class="_ _c"> </span><span class="ff4">[0<span class="_ _b"> </span>p<span class="_ _a"></span>oints]<span class="_"> </span>P<span class="_ _16"></span>ositive<span class="_ _b"> </span>de&#64257;nite<span class="_ _b"> </span>matrices</span></div><div class="t m0 x2 h2 y61 ff2 fs0 fc0 sc0 ls0 ws0">A<span class="_"> </span>matrix<span class="_"> </span><span class="ff7">A<span class="_ _14"> </span><span class="ff8">&#8712;<span class="_ _14"> </span><span class="ff9">R</span></span></span></div><div class="t m0 x45 h4 y62 ffa fs2 fc0 sc0 ls0 ws0">n<span class="ffb">&#215;</span>n</div><div class="t m0 x4e h2 y61 ff2 fs0 fc0 sc0 ls0 ws0">is<span class="_"> </span><span class="ff6">p<span class="_ _16"></span>ositive<span class="_ _0"> </span>semi-de&#64257;nite<span class="_ _b"> </span><span class="ff2">(PSD),<span class="_"> </span>denoted<span class="_"> </span><span class="ff7">A<span class="_ _14"> </span><span class="ff8">&#58903;<span class="_ _14"> </span></span></span>0,<span class="_ _d"> </span>if<span class="_"> </span><span class="ff7">A<span class="_ _14"> </span></span>=<span class="_"> </span><span class="ff7">A</span></span></span></div><div class="t m0 x54 h4 y62 ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x55 h2 y61 ff2 fs0 fc0 sc0 ls0 ws0">and<span class="_"> </span><span class="ff7">x</span></div><div class="t m0 x56 h4 y62 ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x57 h2 y61 ff7 fs0 fc0 sc0 ls0 ws0">Ax<span class="_ _14"> </span><span class="ff8">&#8805;<span class="_ _14"> </span><span class="ff2">0</span></span></div><div class="t m0 x2 h2 y63 ff2 fs0 fc0 sc0 ls0 ws0">for<span class="_ _0"> </span>all<span class="_ _b"> </span><span class="ff7">x<span class="_ _14"> </span><span class="ff8">&#8712;<span class="_ _d"> </span><span class="ff9">R</span></span></span></div><div class="t m0 x58 h4 y64 ffa fs2 fc0 sc0 ls0 ws0">n</div><div class="t m0 x59 h2 y63 ff2 fs0 fc0 sc0 ls0 ws0">.<span class="_ _4"> </span>A<span class="_ _b"> </span>matrix<span class="_ _0"> </span><span class="ff7">A<span class="_ _0"> </span></span>is<span class="_ _b"> </span><span class="ff6">p<span class="_ _16"></span>ositive<span class="_ _b"> </span>de&#64257;nite<span class="ff2">,<span class="_ _0"> </span>denoted<span class="_ _b"> </span><span class="ff7">A<span class="_ _14"> </span><span class="ff8">&#58911;<span class="_ _d"> </span></span></span>0,<span class="_ _0"> </span>if<span class="_ _b"> </span><span class="ff7">A<span class="_ _14"> </span></span>=<span class="_ _d"> </span><span class="ff7">A</span></span></span></div><div class="t m0 x5a h4 y64 ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x5b h2 y63 ff2 fs0 fc0 sc0 ls0 ws0">and<span class="_ _0"> </span><span class="ff7">x</span></div><div class="t m0 x5c h4 y64 ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x50 h2 y63 ff7 fs0 fc0 sc0 ls0 ws0">Ax<span class="_ _14"> </span>&gt;<span class="_ _d"> </span><span class="ff2">0<span class="_ _b"> </span>for</span></div><div class="t m0 x2 h2 y65 ff2 fs0 fc0 sc0 ls0 ws0">all<span class="_"> </span><span class="ff7">x<span class="_ _14"> </span><span class="ff8">6</span></span>=<span class="_"> </span>0,<span class="_ _d"> </span>that<span class="_ _d"> </span>is,<span class="_ _d"> </span>all<span class="_"> </span>non-zero<span class="_ _d"> </span>vectors<span class="_"> </span><span class="ff7">x</span>.<span class="_ _f"> </span>The<span class="_ _d"> </span>simplest<span class="_ _d"> </span>example<span class="_"> </span>of<span class="_ _d"> </span>a<span class="_ _d"> </span>p<span class="_ _a"></span>ositiv<span class="_ _9"></span>e<span class="_"> </span>de&#64257;nite<span class="_ _d"> </span>matrix<span class="_"> </span>is</div><div class="t m0 x2 h2 y66 ff2 fs0 fc0 sc0 ls0 ws0">the<span class="_ _d"> </span>identit<span class="_ _16"></span>y<span class="_ _d"> </span><span class="ff7">I<span class="_ _f"> </span></span>(the<span class="_ _d"> </span>diagonal<span class="_ _d"> </span>matrix<span class="_ _d"> </span>with<span class="_ _0"> </span>1s<span class="_ _d"> </span>on<span class="_ _d"> </span>the<span class="_ _0"> </span>diagonal<span class="_ _d"> </span>and<span class="_ _d"> </span>0s<span class="_ _d"> </span>elsewhere),<span class="_ _0"> </span>which<span class="_"> </span>satis&#64257;es</div><div class="t m0 x2 h8 y67 ff7 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x5d h4 y68 ffa fs2 fc0 sc0 ls0 ws0">T</div><div class="t m0 x5e h2 y67 ff7 fs0 fc0 sc0 ls0 ws0">I<span class="_ _13"></span>x<span class="_ _12"> </span><span class="ff2">=<span class="_"> </span><span class="ff8">k</span></span>x<span class="ff8">k</span></div><div class="t m0 x5f h9 y69 fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x5f h9 y6a fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x60 h2 y67 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x45 h5 y6b ffc fs0 fc0 sc0 ls0 ws0">P</div><div class="t m0 x47 h4 y69 ffa fs2 fc0 sc0 ls0 ws0">n</div><div class="t m0 x47 h9 y6a ffa fs2 fc0 sc0 ls0 ws0">i<span class="fff">=1</span></div><div class="t m0 x61 h8 y67 ff7 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x62 h9 y68 fff fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x62 h4 y6c ffa fs2 fc0 sc0 ls0 ws0">i</div><div class="t m0 x63 h2 y67 ff2 fs0 fc0 sc0 ls0 ws0">.</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>
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