let-u-be-a-square-matrix-with-orthonormal-columns-explain-why-u-is-invertible-mention-the-theorems-y

Name: let u be a square matrix with orthonormal columns explain why u is invertible mention the theorems y
Uploaded: 2020-12-24
Duration: 2 min 27 s
Channel: Numerade
Description: Let $U$ be a square matrix with orthonormal columns. Explain why $U$ is invertible. (Mention the theorems you use.)

Question

Let $U$ be a square matrix with orthonormal columns. Explain why $U$ is invertible. (Mention the theorems you use.)

Oluwadamilola Ameobi · Accepted Answer

So this question says that let you next, you&#x27;ll be a square matrix with Martin. Um Oh, alter normal columns on. Explain why you is invaluable. So if you is a square metrics with Autonoma columns, um and therefore the columns off you would be auto. Go now, because you with the screamer trips with autonomous columns on if the columns of you are to colonel, there must be linear independence nearly independence according to tear Um, for so the columns The columns of you you are auto Go now, Andi linearly independence. But the question says explain why you is invaluable. So since there Linnean since the columns of your Linnean independence on the metrics square there for the metrics has full rank, I would guess that I just square my tricks with four runs or in vegetable according to the vegetable metric story. Um, so since, um, you can say that there, Caesar, Caesar linen, Independence metrics as for around for one. So we can say that you is impossible because the metrics as a full rank Yeah, or we can just say so. Therefore, you&#x27;re s about C. U is equal to one on. Therefore, this implies that using possible I&#x27;m sorry about this so enough for using vegetable

Ashley Boni · Answer

So we have a matrix, you and it&#x27;s an orthogonal matrix. So we know that all columns of this matrix are North Arsenal to each other in pairs. And they&#x27;re also Ortho, Mama. So the norm of each factor a za column is one. And so now we&#x27;re supposing we take a matrix B and we switch up the order of these columns. So it&#x27;s same columns is you, but they&#x27;re in a different order. So ortho or ball banality just means that that that product of each pair of vectors is zero. And the norm is one because the ortho normal. So taking the dot product, um, different pairs of vectors doesn&#x27;t depend on what order they&#x27;re in. So it doesn&#x27;t matter or our columns in. We&#x27;re just We&#x27;re still have the same vectors as columns. So that&#x27;s all that matters. So these air still ortho normal for the normal columns. So V is Orthodox

James Chok · Answer

okay. In this question, we suppose a this grand matrix such that the determined off a Q is equal to zero everyone to explain why a cannot be involved. Well, we know the fact that determinants Oh, a B is equal determinants off, eh? Times by determinants would be This is a fact that we know So we can write is open right tournament or a shoot as being tournament Oh, a times by a sweat, which is tournament off, eh? Sounds like determinants off a square. Then once again, we just a times like tournament. Well, many times, eh? So which is just simply tournament off a times like tournament for a Sounds like 10 minutes. Well, they so which is just determinant. Well, a Q. But we know that this is a criticism to therefore determines for a is equal. She&#x27;s there now remember that a is a convertible. If determinants off is not equal to zero, but because it is a desert, A is not available

Heather Zimmers · Answer

in this problem were asked to explain why we can always find the determinant of a square matrix. So let&#x27;s look at a two by two square matrix just as an easy example. So the elements in every matrix are real numbers. So we have real numbers A, B, C and D, and the determinant is what you get when you multiply the numbers on the main diagonal eight times D. So if you multiply a real number by a real number, you get a real number, and then you subtract the product of the numbers on the other diagonal B times C. So again, we just have a real number minus a real number, and that&#x27;s going to be a real number. So even if it&#x27;s zero, it&#x27;s still a real number. And that&#x27;s really the only expectation for the value of a determinant. It just needs to be a real number. The same would be true for a three by three or a four by four for any size. All you&#x27;re doing is some combination of multiplying, adding and subtracting real numbers, and that will always result in a real number

Angelo Rendina · Answer

So the metrics, eh, is our n by N Matrix and we right this way when we just highlight the columns. So C one is the first column. CTO is the second column and so forth up to CNN, which is the andthe column where the ice call him. He&#x27;s just a column vector. See, I won C i n in our in now. If he right the product, a times a vector x one xn and we expand this product, we see that these equals just x one times the first column, plus x two times the second column, plus all the way to exciting times. The andthe column and these expression here is a vector in a are and and more specifically in the span off the vectors. See one CNN. So now the span of C one CNN is equal toe rn, by definition, even leave every vector me off rn is in the span of C one CNN, but we have served there. We can write on a genetic element of the spawn off. See Juan Si n. It&#x27;s just the metric say that multiplies these nomadic, efficient ce X one exxon. So he&#x27;s is equivalent to saying that the system eh x one xn equals the vector V has solutions the next one accent for every vector V in Oran. But now we know that the system a solutions for every vector we if and only if the medics say is in vertebral as we wanted to see.

Angelo Rendina · Answer

we have the equation A minus. Hey, X Everything Immers equal Ex simmers me Now the first type is to show that bee is in veritable And to do so we multiply from the left by a minus I x so on the left hand side we&#x27;re left with identity and on the right Inside we have a minus a x ex embers Me But now the expression here in blue is by definition the inverse off me and so bees in vegetable. Okay, now that we know this, we can That&#x27;s right. Equation a minus x inverse equal X embers be by inverting both sides to obtain a minor sex equals being birth X. Remember that we invert the product. We get the product off the others in the inverse order. Okay, Now we can move the exxon aside. So let&#x27;s say we write a He&#x27;s equal to a plus being birth X. And now if we show that these metrics in brackets is in vegetable, that we can just multiply by theme verse to have axe equal to something and we&#x27;re done So to do that, we observe here that since a is in vertebral, we can multiply by eating birds from the right to obtain identity is equal to a Los beavers X a m birth. Now these means that X A chambers is the inverse Oh, a plus B in verse which is therefore is in vertebral. So now that we know that we can multiply from the left by the inverse off a plus by beavers And so we see a plus B in verse everything members, they&#x27;re multiplies, eh is equal to X and we have solved for X.

Problem

Let $U$ be an $n \times n$ orthogonal matrix. Sho…

Problem 27 Easy Difficulty

Let $U$ be a square matrix with orthonormal columns. Explain why $U$ is invertible. (Mention the theorems you use.)

Answer

So $U^{T} U=I$ implies that $U$ is invertible.

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

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So $U^{T} U=I$ implies that $U$ is invertible.

Linear Algebra and Its Applications

Lily A.

Heather Z.

Kristen K.

Samuel H.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Lily A.

Heather Z.

Kristen K.

Samuel H.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications