$Determine and compute the elementary matrices: Linear Algebra$

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$Determine and compute the elementary matrices: Linear Algebra$

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Linear Algebra Matrices

Louise W

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Alessandro Iraci

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The elementary matrix associated to a row operation that sends A to A' is the one such that AE = A' or EA = A' or something else entirely?
Matchmaticians

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It's the latter, EA=A'. Here is a nice presentation: https://www.youtube.com/watch?v=gA7m5lttIcU
Alessandro Iraci

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Ok, that's what I deduced from the rest of the text, I wanted to be sure I wasn't messing up, it's been too long since I checked the basic definitions.

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Alessandro Iraci

0

The elementary matrix associated to a row operation that sends A to A' is the one such that AE = A' or EA = A' or something else entirely?
Matchmaticians

0

It's the latter, EA=A'. Here is a nice presentation: https://www.youtube.com/watch?v=gA7m5lttIcU
Alessandro Iraci

0

Ok, that's what I deduced from the rest of the text, I wanted to be sure I wasn't messing up, it's been too long since I checked the basic definitions.

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I assume that the elementary matrices are the ones such that $EA = A'$, that is, they have to be multiplied on the left.

a) We have \[ E_1 = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \qquad \text{ as } \qquad \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 1 & 0 \\ 4 & 9 & 1 \\ 0 & 5 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 2 & 0 \\ 4 & 9 & 1 \\ 0 & 5 & 4 \end{bmatrix}, \] \[ E_2 = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \qquad \text{ as } \qquad \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 2 & 2 & 0 \\ 4 & 9 & 1 \\ 0 & 5 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 2 & 0 \\ 0 & 5 & 1 \\ 0 & 5 & 4 \end{bmatrix}, \]\[ E_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{bmatrix}, \qquad \text{ as } \qquad \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{bmatrix}\begin{bmatrix} 2 & 2 & 0 \\ 4 & 9 & 1 \\ 0 & 5 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 2 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 3 \end{bmatrix}. \]

b) By applying the same elementary operations to the identity matrix, we have \[ E_3 E_2 E_1 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} 2 & 0 & 0 \\ -4 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ -4 & 1 & 0 \\ 4 & -1 & 1 \end{bmatrix}\]

c) To compute the inverse, we apply the inverse operations in inverse order, that is \[ E_1 = \begin{bmatrix} 1/2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad E_2^{-1} = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad E_3^{-1} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \] and so the product is \[ L = \begin{bmatrix} 1/2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 1/2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 1/2 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \]

d) $L$ is already lower triangular and $U$ is already upper triangular, but the canonical decomposition requires $L$ to be unitriangular. We can do so by inserting an extra $E_1 E_1^{-1}$ in the factorisation. So, \[ A = \begin{bmatrix} 1/2 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} 2 & 2 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 3 \end{bmatrix} = \begin{bmatrix} 1/2 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1/2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 2 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 3 \end{bmatrix} \] is the desired decomposition.

Alessandro Iraci

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Louise W

0

I have to show all my work in order to receive full credit. Would writing all of this down be considered good for showing my work do you think? I honestly have no idea because I am behind in the class and therefore very lost.
Alessandro Iraci

0

I think so, I didn't skip a single step. However, two remarks. First understand that I don't know what you did in class, how you are supposed to solve these problems, and which kind of details you are supposed to provide. Second, as a general advice, I strongly recommend you to try to read and understand the solution before submitting it: educators are generally smart, if you submit some work without even understanding what you wrote, they are likely to realise it.
Alessandro Iraci

0

Again, as a general advice, it is okay to ask for help, or even for a full solution, if you don't know how to solve a problem. It is really not okay to passively transcribe a solution that you don't understand. It's actually harmful to your own comprehension and that whoever is grading your assignment can usually tell if you know what you are doing or not, even if you submit a 100% correct solution. I've graded a bunch of assignments myself, trust me on that.
Louise W

0

I guess what I should have said was, is the first one (a) good? I just don't understand how you got E1, E2, and E3.
Louise W

0

You are completely right though. I plan on going back through the textbook and teaching myself everything this week because the exam is Friday. Ahhh I'm wondering if I should turn it in now or just take the 0. You have a very good point
Alessandro Iraci

0

Well, E1, E2, and E3 are the matrices that, when multiplied to the left, correspond to the row operation that you apply to your matrix. I showed all the multiplications, you see, when you multiply E1 to the left to A, you are applying the operation that is to multiply the first row of A by 2. The same goes for E2 and E3, I hope you see it now.
Alessandro Iraci

0

When are you supposed to submit this assignment? My advice is to go through the textbook with the solution in front of you and try to see if you can fully understand it. If you do, go on and submit it. If not, well, you can try, and maybe even get full marks, but expect some questions about it at the oral exam, if you have one. The examiner will want to check if you know what you're doing.
Matchmaticians

0

Indeed the matrices E_1, E_2, and E_3 are obtained from the identity matrix by applying the corresponding elementary row reductions on the identity matrix. That's how you get E_1, E_2, and E_3.

$Determine and compute the elementary matrices: Linear Algebra$

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