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Find an LU factorization of the matrices in Exercises $7-16$ (with $L$ unit lower triangular). Note that MATLAB will usually produce a permuted LU factorization because it uses partial pivoting for numerical accuracy.$$\left[\begin{array}{ll}{6} & {9} \\ {4} & {5}\end{array}\right]$$

$\begin{array}{|c|c|c|c|c|}\hline 6 & {9} & {1} & {0} \\ \hline 4 & {5} & {=} & {\left[\begin{array}{cc}{1} & {0} \\ {\frac{2}{3}} & {1} & {\left[\begin{array}{cc}{6} & {9} \\ {0} & {-1}\end{array}\right]} \\ \hline\end{array}\right.} \\ \hline\end{array}$

Algebra

Chapter 2

Matrix Algebra

Section 5

Matrix Factorizations

Introduction to Matrices

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

04:20

Find an LU factorization o…

03:22

06:49

04:05

03:15

03:28

04:34

04:43

05:35

05:02

Use some form of technolog…

Okay, sir. I have two by two matrix. 6945 Are we here? And I want to find its l u factory ization. Okay, so the way I like to do this is ah, like to start off by writing The Matrix 6945 as the identity matrix times, Some are the matrix itself. So I'm not gonna bother writing that first. So you got the identity matrix here times the matrix itself. Okay, so what we're going to do is be going to perform some role reduction on the second matrix here. Are we going to do it to Echelon for in doing that? This should give us a triangle. A mate on Upper Triangle A matrix has the second Matrix. What should happen then is because we are only getting rid off. We're not getting rid of it, But we making this entry here zero. So when we do that using sort of linear operations, this matrix here should be an upper, as are a lower triangular matrix. Hence, we have our l u R D car factory ization, which should only take us one step. Fortunately, So what is going to happen, sir? once I find the green. Okay, so when we do the reduction on the second matrix Ah, the first row stays as it is. Okay, now, when you multiply matrices starmaker sees if you sort of think off them in in rows, so to speak. So let's say, um a B c D attitude by two matrix and then we have two by two matrix Who's inch? Who's Rose are the vectors, V one and V two. So when we perform this matrix multiplication what we end up with two. The first for it is going to be a V one plus B feature. And the second rower will be C v one plus D v two. But this is sort of like a basic principle off how matrix multiplication sort of works on. This can obviously be extended to arbitrary dimensions of three dimensions or four or five. So this principle that I've written up here so we're using this principle here. So, in other words, the first row off this matrix corresponds to one times the vector 69 plus zero times the vector for five. So, by leaving the first Torosidis the first row off the first matrix stays constant. All right, so what are we doing next? So the next thing we're going to do is we're going to keep the second ror as it is. So that means putting a one over here and we're going to subtract Bye. 2/3 off the first tour. So that road, too, is wrote to minus 2/3 off the first room. So what happens? So four minus 2/3 of six, which is four is zero and five minus 2/3 of nine, which is six is going B minus one. Okay, so what have we done? So if we look at it, let's call this, um, very entry A for now. So you got a time? Six plus one Time zero has to equal to fall. Okay, so six a plus one time zero. That has to equal to four. Furthermore, nine a minus one has two equal five. And if we solve this for a, we should get a equal to 2/3. Sir, if I have used my razor, that's black. I'm gonna do it. Green. This should be 2/3 which is precisely the negative off that. So we do road to plus something than the negative off. That's something girls here in this entry here. So this is not this principle that we've applied here is quite nice. To apply three questions like this, and for future questions, I will probably use something like this as a guide. Although with three by three matrices, the story does get a little more complicated. Eso state Trude for those type of questions. But ultimately, we already have our l u factory ization, which is that Thank you.

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