<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@vertical+block@f8673b9b0215487d835243fd3f650f60" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:OCW+18.06SC+2T2019" data-block-type="vertical" data-has-score="False" data-graded="False" data-request-token="0eca97f8fe4311ee9b8616fff75c5923">
<h2 class="hd hd-2 unit-title">Positive Definite Matrices and Applications</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.06SC+2T2019+type@html+block@fedd7092bd3e4ccda5eb771eb1ffc325">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@html+block@fedd7092bd3e4ccda5eb771eb1ffc325" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:OCW+18.06SC+2T2019" data-block-type="html" data-has-score="False" data-graded="False" data-request-token="0eca97f8fe4311ee9b8616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><img height="240" width="350" src="/assets/courseware/v1/34455a0b0be72a5774ab983db1849f10/asset-v1:OCW+18.06SC+2T2019+type@asset+block/18.06_Unit_III_Overview.jpg" alt="Figure excerpted from 'Introduction to Linear Algebra' by G.S. Strang" /></p>
<p class="caption"><span style="color: #999999;">Positive definite and semidefinite: graphs of x<sup>1</sup>Ax.</span></p>
<p></p>
<p>In this unit we discuss matrices with special properties – symmetric, possibly complex, and positive definite. The central topic of this unit is converting matrices to nice form (diagonal or nearly-diagonal) through multiplication by other matrices. Generally, this process requires some knowledge of the eigenvectors and eigenvalues of the matrix.</p>
<p>Looking for something specific in this course? The <a href="https://mitx.edly.io/courses/course-v1:OCW+18.06sandbox+1T2019/courseware/f7708b4e73004bf9bc9c741f1c19b2e0/38f48933638b403790243d7fca545a9d/1?activate_block_id=block-v1%3AOCW%2B18.06sandbox%2B1T2019%2Btype%40vertical%2Bblock%4051689f6d83d445f2b5ceea8924984b16" target="[object Object]">Resource Index</a> compiles links to most course resources in a single page. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@vertical+block@b2787945c36e4c52a64905c64cb28dce" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:OCW+18.06SC+2T2019" data-block-type="vertical" data-has-score="False" data-graded="False" data-request-token="0eca97f8fe4311ee9b8616fff75c5923">
<h2 class="hd hd-2 unit-title">Further Study</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.06SC+2T2019+type@html+block@61c888a0750649a88a5eccf27c7e1438">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@html+block@61c888a0750649a88a5eccf27c7e1438" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:OCW+18.06SC+2T2019" data-block-type="html" data-has-score="False" data-graded="False" data-request-token="0eca97f8fe4311ee9b8616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>The Java® Demos below were developed by <a href="https://drexel.edu/coas/academics/departments-centers/mathematics/contact/facultydirectory/pavelgrinfeld/" target="[object Object]">Professor Pavel Grinfeld</a> and will be useful for a review of concepts covered throughout this unit.</p>
<hr />
<p></p>
<ul>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/Eigen/" target="[object Object]">Eigenvalues</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/SVD/" target="[object Object]">SVD (Singular Value Decomposition)</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/GaussElim/" target="[object Object]">Gaussian Elimination</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/Determinant/" target="[object Object]">Determinants</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/Gram/" target="[object Object]">Gram-Schmidt = Orthogonalization</a> </li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/InnerProduct/" target="[object Object]">Inner Product of Functions </a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/FourierSeries/" target="[object Object]">Sum of Fourier Series</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/Gibbs/" target="[object Object]">Gibbs Phenomenon</a> </li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/Aliasing/" target="[object Object]">Aliasing</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/ColSpace/" target="[object Object]">Column Spaces</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/LeastSqr/" target="[object Object]">Least Squares</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/javademo/power_method_applet/powerMethod.html" target="[object Object]">Power Method</a> </li>
</ul>
</div>
</div>
</div>
</div>