let-mathbfx-denote-a-vector-that-varies-over-the-columns-of-a-p-times-n-matrix-of-observations-and-l

Question

Let $\mathbf{X}$ denote a vector that varies over the columns of a $p 	imes N$ matrix of observations, and let $P$ be a $p 	imes p$ orthogonal matrix. Show that the change of variable $\mathbf{X}=P \mathbf{Y}$ does not change the total variance of the data. [Hint: By Exercise 11 , it suffices to show that tr $\left(P^{T} S Pight)=\operatorname{tr}(S)$ . Use a property of the trace mentioned in Exercise 25 in Section 5.4 .]

Gideon Idumah · Accepted Answer

So this question when it&#x27;s a note duds. Bye, exercise 11. The change off variables X quarts of p. Why changes the co variance matrix the covert symmetries off the car damages as off eggs in soo. The co variance matrix P Trust falls S p off. Why no recall duds? The total variance off the data. As described by why? Because the governor&#x27;s magics off. Why is Speed trust was SP? So the Toyota Prius would just be the trace off. P. Transports speaking, No. Since P Transpose s meth lab you similar Sue s? It implies that the off the same truth that sees truth off s is the sentinels trees off P transpose SP. I can get this if you look at what&#x27;s by exercise trying to five in section 5.4. So since the actress is the same thing on, then recall that&#x27;s we call duds. Total variance is just some of truth is just a song off the trees. This implies that the total variance Dr He&#x27;s on changed because the tress are both the same by the change off Vary booze, eggs, quarts of what

Jack Chen · Answer

Okay, so for this problem, part of one we know that X one up to eggs in Hi. In, Um I mean deviation for that means, um X one process x two up to eggs. Then we will be, ah zero vector which is and by 10 Victor Lemmens If we defied of extra top very close to one and by one vector with all with all inches are one We have a magics modification x one x two Exane multiplied by w equals 20 which we secretly into this condition. But this muse p transpose times um, this matrix multiplication because with zero. But by our definition, patrons post times on this metrics will be why one up to y in So basically change the the order off multiplication and that this line means why one upto y in How are also in a media in deviation form. So this is the proof for part one. And for part two, we know that there are coherence. Metrics for X well is defined in this way and the full for why the cover and men covers metrics s why he closed toe one over a minus one. Why times why transpose and by the definition of why, equals two p transposed eggs times teachers, post eggs and the amateurs supposed outside in a barrel off veterans posts. We have X transpose times p and that we we know that X choose post times extents extras post divided by impound and minus one equals two ways. So in the end we have Peter&#x27;s Post Times, Ace&#x27;s Times Peep, which proof the which proofs The second statement.

Jack Chen · Answer

Okay, so for this properly if we let b equals two x one head up to aches and head are which heart, which is Ah, meat mean division form where X I had equals told young you know, vector X I minus the mean vector. So the covenants metrics as it is defined by one over a minus one B times be transposed, which is one number a minus one tons Summation iPhone 12 in eggs. I had times eggs I had chest pose in the by the definition off X I head, We have the following expression so x I minus him times x I miners in chills Post which proofs which have this phone has this seeming there, um is has a similar form to the sample variance which proofs on the statement in this problem.

Angelo Rendina · Answer

That&#x27;s right, The medics of the coefficients Big Axe as the first column is all ones and the second column has the axes of the points X one Exxon. And we know that some of X, which is what I did Note X one plus plus Exxon and zero. So the point starting mean deviation form We want to show that Well, X transpose X is a dagger on metrics. So let me call the first column of X C one and the second column of Ecstasy too. Now X transpose x Well, that&#x27;s see one transport see to transpose. But as Rose and then C wants you to thought, these are two by two matrix and well, we have C one transports you want you want transparency to and then see to transport C one and C to transport, see to All right, no c one Transparency One is just well, some of ones. And we have exactly in data points. So it&#x27;s n them for C one trust. Beyonc? We see that that the some off ex multiply by one so there&#x27;s just some of X and similarly c two transposed. The one is some of ex again and then see to transport see to it the sum of the squares. So some X squared. But now we&#x27;re done because we know that some X zero. So we have an 00 sum of X squared, which means that X transpose x. He&#x27;s a diamond.

Runpeng Li · Answer

All right. So in this problem, work even for skipping a, uh, a quadratic a form that is ny uh, I x one squared us seven x two squared us 11 x three squared, minus eight. Ex one next to and plus eight. Excellent. Next three. So the pristine we need to do is to write out the matrix bye, which is a three by three matrix. Uh, so So the way we need to, uh, right out, right how the Matrix A is to compare the coefficient of the quadratic born to our general coded form for three by three matrix, which we have already done in the previous work. So I would not go through the approach here, so I&#x27;ll just write down the matrix a year. So it&#x27;s nine naked four or and next, four, seven, zero, and or zero 11. Okay, So now the next step is to find out the wagon vectors corresponding to the given Aiken value. So? So to do that, the approach that I would recommend to do it&#x27;s to consider linear system A. My list loved I because they&#x27;re so by that way, um, we just do the reduced to this system and find out a solution. So I won&#x27;t. I will write down the pros. The process here and because it&#x27;s just a rule, reduced reduction won&#x27;t be too hard and will be a mine. It&#x27;s not that I was zero. And turns out we have three again vectors for a second doctor. ISS effective to give to one with again value love. No one wishes three and second of Victor is like to 1/2 and one and one and which, which correspond to the vector. Sorry. Uh, Lambda too. That is night also. Half we&#x27;d be three. We should call respond to, uh, one negative 1/2 of one and correspondent, which corresponds to London. Three. That is 15. So Okay, so right now we recall the&#x27;re, um uh, This The room is given our given by our textbook, so I&#x27;ll just write down here we have. We have a symmetric and each idea values are distinct. Distinct. Then the Eigen vectors. Our fourth Avenue or something. Okay, So that means Thea Aiken vectors. We get these three back wagon victors are all definitely view. Uh, you want you can just check whether their authority this theory tells us that, um it must be or, you know, so we can just write down the matrix p directly by these. These three factors Just negative too. Give to one. And next. One, three. Sorry. 91 two, two. Oh, sorry. Uh, so remember to just a right down the matrix on the ortho one matrix in, uh, normal lines to form. So I&#x27;ll just directly right down it all my life form. By my calculation, that would be, uh, 1/3 which is still scaler. We have nothing negative to negative too one and elected. One, two, two. And the last column is too active. One. I do. So this is our or a token of matrix speak. So the last thing we need to d&#x27;oh is too. Find the change of variable X equals P y transforms X t x transpose X transpose a axe into a quadratic form with no course product term. So this by we can just write down. There&#x27;s write down the correct equaled uniform directly by our theory. Um, in the textbook, it&#x27;s just Ah, the Eiken values off, eh? Times. Sorry. Uh, unbearable should be. Why, Yeah, I can values number one why one squared so that, uh, lambda and times why ends where where off Ramdas are I can values. I can&#x27;t bel us off a and the work even the Yangon medics off a to B three Say it&#x27;s, uh, 39 15. So we will just write down directly. That will be three. Why one&#x27;s weird? I my two squared and 15 Why three squared. So where?

Alayna Abraham · Answer

So let&#x27;s try and find the size of P. So that&#x27;s our first part of this problem. And we&#x27;ve been given two pieces of information. What P is equal to and what how are a matrix is to find us. So are a matrix is sometimes, and and R P is equal to a times a transposed times A I&#x27;ll take in the embers off times a transposed again. So it&#x27;s important to know that when you transpose a you&#x27;re switching the rows and columns. So now 80 is now M. Times M, and when you multiply your T and a together, you are multiplying and times and I m tight. And when you do that, you get in time. Ben. Now, when you take the inverse of this hoping because it&#x27;s end times, then it will also be in times that again. So let&#x27;s put in what we know into this formula, so P is equal to and times in times end times and times and times him. There&#x27;s a lot of Eminem&#x27;s, so when you take the multiplication of the first part and you multiply and we know that our rows and columns are equal to each other, e wish there was other variables I could tell you but the admin and are equal to each other. I&#x27;m sorry. So our m times heart on and and we&#x27;re multiplying that all by end times, um, and again we see that beast, You are equal to each other so we can multiply that out. So now P is equal toe sometimes m and that&#x27;s the size of a pea. Now, the second part is and so this is two different parts p times a equal a. Now, let&#x27;s do that first so we know what P is on. So we know that the formula for key is and we haven&#x27;t above as being a times a transpose times a and the inverse of all that times a of tea. And because we&#x27;re doing p times a we have add another A in here, Um, we know because of above that this is essentially and I didn t matrix so we can define this part is being pie and this being are a that we multiply. So then a times an identity matrix And this is a rule that you should know anything times and identity matrix is always gonna be that matrix. So it&#x27;s gonna be a so does p A equal A yes. And the second part he squared equal to P. So he squared is just p times p So I will just rewrite our formula again twice because we are squaring and I&#x27;m writing the whole thing out And if you remember above said that p is an m times m make sure x and that&#x27;s essentially identity. So this part is I. And when you do that and you multiply I all the way through we distribute the I all the way through this or I&#x27;m sorry when we put I into this substitute I in to this&#x27;ll again remember above anything well supplied by an identity matrix is that thing. So all right, answer is our formula for P so that people to be this is p squared is equal to p. Yes, it does. And I Last thing to do is to prove that P is a symmetric matrix. And so if we want to prove that we need a pro of the beauty isn&#x27;t able to pee because if you and no, there&#x27;s this rule with multiplying anything with the transport, so a matrix times be matrix and the whole thing transposed is essentially equal to be transpose times a transpose. So if it we want to prove that p a t is a symmetric matrix, we first need to prove that a tee times a business symmetric matrix. So we have to take the transpose of the whole thing And when you dio t And if you transpose on already transposed that just becomes a So this is our original. Therefore, it is transport symmetric matrix and a negative one. This is already symmetric So then a t a a and the whole thing is already taken The inverse of the whole thing is already gonna make it symmetric. So then now, because we&#x27;re not looking for P you were looking for P of tea. So let&#x27;s transpose the whole formula given what will you know so way have a transposed and the whole thing transposed and again that will you get rid of than a t a couple days in versed in mass transpose and a T And this is gone. This will essentially just be the same thing. So when you&#x27;re done, you will just get a times a transposed times a day, and the whole thing is in an inverse times, a transposed. And if we remember about that is our original P formula. So then p of t is equal to P, and that makes into a symmetric matrix.

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The sample covariance matrix is a generalization …

Problem 12 Easy Difficulty

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

Linear Algebra and Its Applications

Anna Marie V.

Alayna H.

Kayleah T.

Caleb E.

Absolute Value - Example 1

Absolute Value - Example 2

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Linear Algebra and Its Applications

Anna Marie V.

Alayna H.

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Caleb E.

Absolute Value - Example 1

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

Anna Marie V.

Alayna H.

Kayleah T.

Caleb E.

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

Linear Algebra and Its Applications