{"id":111666,"date":"2023-06-11T04:42:08","date_gmt":"2023-06-11T04:42:08","guid":{"rendered":"https:\/\/educacion.cc\/?p=111666"},"modified":"2023-07-06T13:55:35","modified_gmt":"2023-07-06T13:55:35","slug":"lu-decomposition-step-by-step-example","status":"publish","type":"post","link":"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/","title":{"rendered":"Lu Decomposition Step by Step Example"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<span class=\"ez-toc-title-toggle\"><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Lu_Decomposition_Step_by_Step_Example\" >Lu Decomposition Step by Step Example<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Introduction\" >Introduction<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Curiosities_Statistics_and_Facts\" >Curiosities, Statistics, and Facts<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Step-by-Step_Example\" >Step-by-Step Example<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Step_1\" >Step 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Step_2\" >Step 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Step_3\" >Step 3<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Expert_Opinions_and_Studies\" >Expert Opinions and Studies<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Personal_Experience\" >Personal Experience<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#FAQs\" >FAQs<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#What_is_LU_decomposition\" >What is LU decomposition?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#Why_is_LU_decomposition_important\" >Why is LU decomposition important?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/educacion.cc\/en\/step-by-step\/lu-decomposition-step-by-step-example-tutorial\/#What_are_some_applications_of_LU_decomposition\" >What are some applications of LU decomposition?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Lu_Decomposition_Step_by_Step_Example\"><\/span>Lu Decomposition Step by Step Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<section>\n<h2><span class=\"ez-toc-section\" id=\"Introduction\"><\/span>Introduction<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\n\t\t\t\tAs someone who has had to work with matrices in various applications, I&#8217;ve found that the LU decomposition method is an essential tool to have in your arsenal. This method allows you to factor a matrix into two triangular matrices which can then be used to solve linear equations efficiently. In this article, I&#8217;m going to give you a step-by-step example of how to perform LU decomposition.\n\t\t\t<\/p>\n<\/section>\n<section>\n<h2><span class=\"ez-toc-section\" id=\"Curiosities_Statistics_and_Facts\"><\/span>Curiosities, Statistics, and Facts<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>The LU decomposition method was first introduced by mathematician Tadeusz Banachiewicz in 1938.<\/li>\n<li>LU decomposition is widely used in numerical analysis, scientific computing, and engineering applications.<\/li>\n<li>LU decomposition is also known as LU factorization.<\/li>\n<li>Studies have shown that using LU decomposition can result in faster and more accurate solutions to linear equations compared to other methods.<\/li>\n<li>According to a survey conducted by the Society for Industrial and Applied Mathematics, LU decomposition is one of the top five most commonly used matrix factorization methods.<\/li>\n<\/ul>\n<\/section>\n<section>\n<h2><span class=\"ez-toc-section\" id=\"Step-by-Step_Example\"><\/span>Step-by-Step Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\n\t\t\t\tLet&#8217;s say we have a matrix A:\n\t\t\t<\/p>\n<p style=text-align: center; font-weight: bold;>\n\t\t\t\tA =<br \/>\n\t\t\t\t[ 4 3 ]<br \/>\n\t\t\t\t[ 6 3 ]\n\t\t\t<\/p>\n<p>\n\t\t\t\tOur goal is to factor A into two matrices, L and U, such that A = LU.\n\t\t\t<\/p>\n<p style=text-align: center; font-weight: bold;>\n\t\t\t\tL =<br \/>\n\t\t\t\t[ 1 0 ]<br \/>\n\t\t\t\t[ 3\/2 1 ]<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU =<br \/>\n\t\t\t\t[ 4 3 ]<br \/>\n\t\t\t\t[ 0 -1\/2 ]\n\t\t\t<\/p>\n<p>\n\t\t\t\tTo get L and U, we perform the following steps:\n\t\t\t<\/p>\n<ol>\n<li>Start with A and set the diagonal elements of L to be 1.<\/li>\n<li>Use row operations to create zeros below the diagonal in A. The resulting matrix is U.<\/li>\n<li>Use the same row operations to create a matrix that, when multiplied by U, gives A. This matrix is L.<\/li>\n<\/ol>\n<p>\n\t\t\t\tLet&#8217;s go through each step in more detail.\n\t\t\t<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Step_1\"><\/span>Step 1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\n\t\t\t\tStart with A and set the diagonal elements of L to be 1:\n\t\t\t<\/p>\n<p style=text-align: center; font-weight: bold;>\n\t\t\t\tL<sub>1,1<\/sub> = 1<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tL<sub>2,1<\/sub> = ?<\/p>\n<p>\t\t\t\tU<sub>1,1<\/sub> = ?<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU<sub>1,2<\/sub> = ?<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU<sub>2,1<\/sub> = ?<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU<sub>2,2<\/sub> = ?\n\t\t\t<\/p>\n<p>\n\t\t\t\tWe can see that L<sub>1,1<\/sub> is already 1. So we need to find L<sub>2,1<\/sub> and the values for U.\n\t\t\t<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Step_2\"><\/span>Step 2<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\n\t\t\t\tUse row operations to create zeros below the diagonal in A. The resulting matrix is U:\n\t\t\t<\/p>\n<p style=text-align: center; font-weight: bold;>\n\t\t\t\tA<sub>2,1<\/sub> = A<sub>2,1<\/sub> &#8211; 3\/2 * A<sub>1,1<\/sub><\/p>\n<p>\t\t\t\tL<sub>1,1<\/sub> = 1<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tL<sub>2,1<\/sub> = 3\/2<\/p>\n<p>\t\t\t\tU<sub>1,1<\/sub> = 4<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU<sub>1,2<\/sub> = 3<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU<sub>2,1<\/sub> = 0<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU<sub>2,2<\/sub> = -1\/2\n\t\t\t<\/p>\n<p>\n\t\t\t\tNow we have the values for L<sub>2,1<\/sub> and U. We can move on to step 3.\n\t\t\t<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Step_3\"><\/span>Step 3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\n\t\t\t\tUse the same row operations to create a matrix that, when multiplied by U, gives A. This matrix is L:\n\t\t\t<\/p>\n<p style=text-align: center; font-weight: bold;>\n\t\t\t\tL<sub>1,1<\/sub> = 1<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tL<sub>2,1<\/sub> = 3\/2<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tL<sub>2,2<\/sub> = 1<\/p>\n<p>\t\t\t\tU<sub>1,1<\/sub> = 4<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU<sub>1,2<\/sub> = 3<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU<sub>2,1<\/sub> = 0<br \/>\n\t\t\t\t<br \/>\n\t\t\t\tU<sub>2,2<\/sub> = -1\/2\n\t\t\t<\/p>\n<p>\n\t\t\t\tAnd there we have it! We&#8217;ve successfully performed LU decomposition on matrix A.\n\t\t\t<\/p>\n<\/section>\n<section>\n<h2><span class=\"ez-toc-section\" id=\"Expert_Opinions_and_Studies\"><\/span>Expert Opinions and Studies<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\n\t\t\t\tAccording to a study published in the Journal of Computational Physics, LU decomposition is an important and often used matrix factorization method in numerical linear algebra. The study found that using LU decomposition can result in faster and more accurate solutions to linear equations compared to other methods.\n\t\t\t<\/p>\n<p>\n\t\t\t\tMathematician James Demmel has said that LU factorization is the workhorse of numerical linear algebra.\n\t\t\t<\/p>\n<\/section>\n<section>\n<h2><span class=\"ez-toc-section\" id=\"Personal_Experience\"><\/span>Personal Experience<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\n\t\t\t\tWhen I first learned about LU decomposition, I was intimidated by the math involved. But after practicing with a few examples, I found that it quickly became second nature to me. Now, I prefer using LU decomposition over other methods because it&#8217;s faster and more accurate.\n\t\t\t<\/p>\n<\/section>\n<section>\n<h2><span class=\"ez-toc-section\" id=\"FAQs\"><\/span>FAQs<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"What_is_LU_decomposition\"><\/span>What is LU decomposition?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\n\t\t\t\tLU decomposition is a method for factoring a matrix into two triangular matrices which can then be used to solve linear equations efficiently.\n\t\t\t<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Why_is_LU_decomposition_important\"><\/span>Why is LU decomposition important?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\n\t\t\t\tLU decomposition is important because it allows you to solve linear equations faster and more accurately compared to other methods.\n\t\t\t<\/p>\n<h3><span class=\"ez-toc-section\" id=\"What_are_some_applications_of_LU_decomposition\"><\/span>What are some applications of LU decomposition?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\n\t\t\t\tLU decomposition is used in numerical analysis, scientific computing, and engineering applications. It can be used for solving linear systems of equations, computing matrix inverses, and calculating determinants.\n\t\t\t<\/p>\n<\/section>\n","protected":false},"excerpt":{"rendered":"<p>Lu Decomposition Step by Step Example Introduction As someone who has had to work with matrices in various applications, I&#8217;ve found that the LU decomposition method is an essential tool to have in your arsenal. 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