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[ 10 2 12 3 11 1 -3]Apply Householder Transformation to reduce A to a tridiagonalmatrix...

Question

[ 10 2 12 3 11 1 -3]Apply Householder Transformation to reduce A to a tridiagonalmatrix

[ 10 2 1 2 3 1 1 1 -3] Apply Householder Transformation to reduce A to a tridiagonal matrix



Answers

Use row operations to change each matrix to reduced form. $$ \left[\begin{array}{ll|r} 1 & 2 & -1 \\ 0 & 1 & 3 \end{array}\right] $$

Get this matrix and reduced form. So let's first focus on our patient left most non zeros to be one road to this three. We can divide road to buy three way. We can say that as we want to take one third of road to and put this into road to meeting Rose one in three will remain unchanged in row two is going to be a third of what it originally is. So a third of zero, a third of three, a third of negative six and a third of one. Mhm. No. Um What we can do next is so let's um let's replace row one by taking row one and subtracting two times, wrote to to it. Sure. So rose to and three will remain unchanged and row one we'll get a zero above this one is what we're trying to get. So we take real one subtract two times road to one minus two times zero as one, two minus two times 10 negative two minus two times negative two. That we negative two plus four which is positive too And then negative 1 -2 times one third. That would be negative one minus two thirds or negative three thirds minus two thirds which is negative five thirds. Okay now we could um try to get a zero in row three by just adding rows three and two together. In writing this in row three. So Rows one and 2 Remain unchanged in row three is found by adding rows two and three together. So zero plus 001 plus negative one and zero negative two plus 201 3rd plus negative one third zero. We're not only reduced form because I left most are one are we have a row of all zeros which is at the bottom, we have zeros above and below. Our luck most ones. So this is a reduced form.

Okay, we give this augmented matrix on, we'd like to put it into reduced wrote. Excellent form. So the first thing I'm gonna do is well, we want to get this to be our pivot for the rest. Get rid of this one for this. So clearly, we just have to add grow 1203 So we'll do. You ready? Three plus row one. And that's gonna replace right three. Okay, so let's do that here. So spread one will stay to say nice one one, right to a state of same 01 minus one. That right, three. Now we'll have one post. My response would be zero minus one was once we zero, Adam will have one plus one. It's gonna be on if we do our thinking. We're gonna have a, uh minus one minus one. They could stay the same, but we will have one. Thats minus one, which is gonna give us zero. Right. Okay, so, uh, that's half our eyes do half off. Right? Three announced. That would get us to a one so that we're in reduce restaurant for so we'll have right half a break. Three. Replacing right. Three. Shut my window. Some guy was met by, of course, have. So, uh, everything stays the same. Modest. 11 minus one minus 10 But this becomes a one now, this one one on. See? Right, 00 Okay, so now we're gonna have to make these zeros above up. It's so clearly we're gonna have to do a row. One is gonna be replaced with row one minus row three. Andi Road to is going to be placed with row two. Plus right, three. Okay. Said we'll debate for days now. Um, 1st 2 entries of Rwanda gonna stay to say, and then we have one minus one, which could be zero and then more. That's what my Syria is just going to stare minus one for the second round. Will have 01 They say ineffective. But there will be adding one to this so that I'll be zero on the last one is going to say I have 0010 Okay, So anything that is, get rid of analysis minus one here. So clearly, we're gonna have to have replaced row one with row one. Plus wrote to because we need foot. It's fearing. Reduced rates from for miniseries, but all of our events. So we conclude that's gonna work out up. Definitely changed. That's become zero. That's zero of minus one plus minus one. So that's gonna be mice, too? Yeah. And then second row is unaffected. And same with the third race. So this should be augmented Matrix in work reduced Brooks on four.

I came to give this orbited matrix on, we'd like to put it into reduce row. Excellent form. So first, let me take. Probably gonna be saving us a lot time if we replace, uh shall we say, row three with no one. So, actually, no, we'll want to dio I want to do. That's just Teoh, Replace Roti with the row one. See what we end up with her. So one minus 11 on this will pay one. We prefer one with Texas is good. Eat minus one. Now, this will be one on this will be minus Well on. This is gonna be zero, uh, minus warm. One on this is gonna be one minus one minus. Why? Hey. So, um, if we do, um, row one minus wrote to We should get rid of both of our zero says we have ones. The reserves of stock generative. This one he upset will be replacing row one with red one bites right to. So we'll have, uh, 100 on minus what? Minus ones. Goody minus two. I would be like zero minus 11 one one, minus one. My squad. Okay. Next up, we're gonna be one of you placing road three with Roe three minus for at once so that we get rid of this could also want him to be here. Say we will be doing rope one. It's very three, maestro. One is equal to equal to is gonna replace growth three. I say we have 100 to. And you know, I'm also gonna make road to equal Teoh replaced with what? Minus right to is gonna replace right so that we have a one here said we've got 01 minus one, minus one on. Uh, Row three is gonna have this one minus that one since 20 This is good. Minus one. This bit will be mice want as well on December is gonna be minus one minus minus my scrub. My past is its be minus one plus two. So this could end up one here yet? So nice that we're gonna get you want to get rid of their suit clean. I'm gonna have to do Road three. Add grow to ongoing replacing, right? Three with that. So we'll have 100 to, uh, 01 minus one. My one. And then here we have 00 minus one of my spiced meat minus two on day one. Ad, uh, minus one while that's cool, baby. Zero row three, we're gonna want toe Do minus 1/2 are very three replacing the three so that we've got off as one. So have 10001 minus one 001 Sierra minus 12 on to get rid of the cleanest zeros. But wall a pivot with really good after a place wrote to with ready to plus road three on Since there's zero on the end bit always gonna happen is that minds one comes here I said tu minus one on this should be row Matrix in reduced wrote a long form

Let's use our operations to transform this matrix ro reduce, uh, slime. For now, there are different ways to do this. I'm just going to provide 1/2 to get there. So we first want to focus on getting a leading one in Row one. So let's switch our row one and to or round doing that, we'll have rolled. One will not be one negative to three one in road to will be three negative 112 in Row three is being left alone. Next, let's work on getting zeros below the leading one wrote one. So what if we take row two and subtract three times row one from it and put that in the road to then Row one will be left alone in Row three will be left alone and we'll take from above road to minus three times. Row one. So we have three minus three times. One that's three minus three or zero in negative. Two minus three times. Negative too, would be go away. Uh, Rocha. Negative. One minus three times. Row one. It's fine. Um, one minus three times there is one minus nine, which is negative. Eight and two minus three times one is two minus three, which is negative one. Let's now take Roath re from above and subtract two times, wrote one and then put that in our new row three. So from above rose one and two remain untouched. And we'll do wrote three, which is two minus two times. Row wants to minus. Two times one is two minus two or a zero one minus two times negative, too. Is five negative three minus two times three. This negative nine in negative one. My s two times one. It's negative. Three. Next. Let's get ah leading one wrote 11 way we could do this is to just take 1/5 of row two. You replace wrote you with it. Then we'll have 1/5 of 0 1/5 of five negative 8/5 negative. 1/5 rose. One in three will stay the same. Let's get zeros below the leading one and wrote shoes to do that. We could take, um, wrote three from above and subtract five times Where to? So zero minus five times zero is 05 minus five times one is zero negative. Dine my ass five times. Negative. 8/5 um iss. Let's see. Negative nine minus five times negative. 8/5. Negative one and then negative three minus five times is negative. Two. We'll keep Rose one into the same. Now we want to get a leading one in Row three. And to do that weaken. Just take the opposite side over. 03 multiplied by negative. One 0012 Now now let's get zeros above the leading one in a row. Three. We can do this by taking road to for above and adding a fix of Row three to it. Including that road to so wrote two times 8/5 of growth three zero plus APIs of zero is zero one plus eight. That zero is one negative 8/5 plus 8/5 of +10 in the negative. 1/5 plus 8/5 of to is three, um, one in three. But say now, let's get um zero in row one by taking row one from above and subtracting three times. Wrote three, replacing road one with it. So one minus three times zero was one negative. Two minus three times zero is negative. Two three minus three times one is 01 minus three times to his negative five. Next up, trying to get a zero above the one. We can do this by taking row one last two times wrote to putting that in a row one. So Rome one one plus two times here was one negative two plus two times one is zero zero plus. Two times there was zero negative. Five plus two times three is negative, five plus six, which is one in row two and three staying the same. And we are done because we have the three by three and then any major eggs on the left, but


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