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Spectral Data for B problemot04HAR...

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Spectral Data for B problemot04HAR

Spectral Data for B problem ot04 HAR



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Construct a spectral decomposition of $A$ from Example 2

So we need to find the spectral decomposition of matrix A, which is equal. Yeah, right. Speaks minus two during this one, Mhm. And then minus two. Yeah. Six. Yeah, minus one. Mhm. And then minus one. My next one. And finally. So this is given medics Specter, the composition is to be found. Now we know that we can find the Eigen Values and Eigen vectors of this, uh, equal this matrix by using the characteristic equation. So by using the characteristic equation, we can find the Eigen values and the Eigen values are I'm the one in quantum eight. Yeah, And then Lambda do but equals to six. Mhm, mhm, mhm. And then number three equal to three. Yeah! Mhm number three equal to three. These are the three I can values that we can find by using the characteristic equation. Now. Now, what we can do is we can find the Eigen vectors by using this by putting these by Eigen values and the characteristic equation obtained. And then we get the corresponding Eigen vectors. Let x one b the Eigen vector for Lambda One. It takes to be the Eigen vector for Lambda too. And it extremely the Eigen vector for the Lambda Three. So we get X one. Yeah. Equal system. Yeah, it's one equals two minus 110 Sure. Thank you. And then next week or two. Yeah. Okay. The yeah. Extra quarter minus one minus 12 Mhm. Yeah. And then next, Three quarter. Mhm. Mhm! Yeah, yeah, yeah, yeah. 11 and one. Yeah! Mhm. Mhm! And then we as we got And then as we got these Eigen Eigen vectors for the corresponding Eigen values let you win u two and u three b the normalization matters of X one, x two and X three, respectively. Now, therefore, we can normalize. Excellent. And then get you will. As you know, one equals two. Mm minus one viral too. 102 and zero. Mhm! Yeah, yeah. Mhm. Yeah. Seven Little League. By normalizing the other two icon vectors, we get you to equals two By normalizing X two, we get you to equals two. No! Yeah, minus 106 Mhm. Yeah. Minus 106 and two pair of six. So minus one bedroom, six and two bedroom six. We are normalising Eigen vectors. Mhm to obtain this. You want you do. And you three similarly, this minus 16 to minus one Battle 16, 456 And then you get you three equals two. Yeah. Mhm. 1031 by road three and 123 Mhm. Mhm. Yeah. Mm, yeah. Mm. No Knees under nice Eigen vectors. No. If we can combine human, you do. And you three we will obtain, uh, we're all obtaining normalized like, uh, matrix, which is which is made up of normalized, like in Victor's, which we can call us in. So n equals two. It's basically clubbing. You want you to and you tree. So we get an equals two minus one. Little too minus one. Beirut to one by route two and zero. Mm hmm. Yeah. The mhm. Yeah, yeah, yeah. And then minus one. Made up six. Yeah. Mm. Mhm. Mhm. Minus one by six. And divide objects. Yeah. Mm. Yeah. And one barrel, 31 barrel, three and one barrel. So basically, we're just clubbing up human u two and u three. Yeah, Yeah, yeah, yeah, Mhm, Yeah. Now we got in. Now we can Now we know that a equals two now since we found And we also find we found out the Eigen values and Eigen vectors so we can get the diagonal matrix D as but D equals two. Yeah. 800 The Yeah. 060 Yeah. Yeah. And 003 Basically, using the Eigen values that we found out. I just wanted to say that we can also find these. Okay. Yeah, yeah, yeah. So this is which we can also find using the Eigen values, which we found out earlier. Now what we need is the spectral decomposition, which is equals two. Yeah, Yeah. London one into. Yeah. You want to Do you want transport? Yeah, Yeah, yeah, yes, yeah. Under the window. Mhm. Yeah. You to enter you to transport? Mhm. Yeah. Mhm. Mm. Yes. Yeah. I'm not three in two. Yeah, U three in two or three transports. Yeah. Okay. Yeah. Yeah. So we can multiply. You want to You want transport? You go into YouTube transport and you three and two u three Transport transport. And since we know that Lambda one equals to eight, Lambda two equals to six and Lambda three equals to three after multiplying. You want to You want transport you to into YouTube transport. And you three to you three Transport. We finally get the value of a as equals two, eight times off my path. Minus half zero. Uh huh. Minus half. Yeah. Zero. And then minus half. Half and zero. Yeah, And the third row is 00 and zero. Mhm. Yeah, Yeah. Mm. Class lambda. Two times of Lambda Tau is six. So there were six times over the Sure one by six. One by six and minus two by six. Yeah. Yeah. Okay. Yeah. Mhm. Mhm. Yeah. One by 61 by six and minus two by six. Once again, the second rule is the same as the first rule. This is you two times you to transpose. Yeah, yeah, yeah! Mhm, Yeah! Yeah! And then minus two by six minus two by six and four by six. Mm, yeah, yeah. Mhm. Yeah, Yeah. Uh huh. 456 which is two by three. But I let it be like that. And then we can also find you three times you three transports and write it as Lambda three times you three and two U three Transport, which is, as we know under three equals to treat. So therefore, three times off, one might be all the all the elements of this matrix you three times you three transports are won by three. So therefore plus three times off. One by three, one by three, one by three. Okay, mhm. And then one by three Yeah, yeah, One by three Yeah. One by three Yeah. Mhm one by three Yeah, yeah, yeah, On my three and one day Yeah, yeah, yeah! Mhm! So this is a spectral decomposition of the Cuban matrix A And this is this is eight is equal to Lambda one, This matrix, this whole matrix is You want times you want transposed and plus six times, which is Lambda two times you to interview to transport, which is these metrics and press Lambda three, which is equal to three times new three into you treat transport, which we obtain as this. So this is a spectral the composition of the given matrix A which is equal to six minus two minus one minus 26 minus one minus one minus one in five. Thank you

We need to find the specter the composition of the given metrics Gay, which is equal to six. My Mr minus one My mystery six minus one. Find this one on this one and site. This is the Matrix, which is given who expected The composition is to be found now. You know that we can use the characteristic question minus Nanda is determining to reminder. I can values a minus number equal to zero. We can find three Eigen values and those are using the characteristic impression those are Nanda. One equals to aid number two equals 26 and number three equals 23 No, there let excellent next door and x ray b the Eigen vectors of the Eigen values. Remember one number two and then battery perspective. So therefore X one equals two minus one. Mhm one. It's you know, you can find the Eigen vectors by substituting the Eigen values in the characteristic equation which is obtained. And then X two equals two minus one minus one and two and then x three, which is Ivan Well again, Victor, corresponding to the third eye can value that is three equals to 11 and one now that you want and you three let you want to do and you see the the normalized Eigen vectors of excellent, extremely extreme, respectively. Therefore, we can obtain with proven equals two minus one by Ruto and then minus four plus one and zero This is the normalized convicted for icon Vector X one Similarly, for ICANN vector extra we can obtain. The normal is I can vector Else you do equals two minus one by Route six minus 106 and two by Route six And then you three calls to for Eigen Vector extreme. The normally second victories one by routine one by routine one by one. I'm sorry. Yeah, when they look at you and they can when they're upset for these are the three normalized vectors. Now we can club human U two and u three. That is all the normalization vectors as one matrix, which is in equals. Two you want as the first column you two as the second column and you three is the third column therefore minus one by two 12 zero. This is human. The first call U two s the second one so minus one by rope six and then minus one right hook six and then provide would fix. And then in the third column, we can put you three, which is one bare root tree, one directly. Anyone battle. This is the and metrics, which we cortex clothing the three normalize Eigen vectors of the respective Eigen vectors. It's one extra, an extra No, we know that easy for the spectral decomposition equals two. The under one into Cuban. Do you want transport Question number two and told you into local transport This land battery interview tree into a three transfers? No. Therefore we can find the spectral decomposition by finding the values of you want into your own transport and then multiplying it by land and then adding the product of number one number two transport and then multiplying trial under two and similarly number three into them nutrients so we can find you land a one into human Inter human transport equals two, eight and two. My next one Very cool one very cool and zero minus Umberto. Then one by Rocco and zero. So this is You want times you want transports, which is minus 102 So this is minus one that, of course, one bottle. And you know, this is you want to impose. So which is according the painters? Uh huh. Than minus all. And then needle minus off class zero and then zero zero and zero. Similarly, we can find lumber. You go into your principles and then you three into your country. And then finally we get the structure. The composition of the government s A equals two eight times off, minus half zero and then minus half, half zero and then zero. Mhm needle plus 16, 12, one by six one basics. I missed two by six. One by 61 by six and again minus two by six. Then minus two by six minus two by six. And then four by six. Yes, lander. 3 to 3 to three times sports just three times. Which is the number three you do you treat France course product, which is one battery, one battery one by three. One day three, one mighty And by three, one by three, one by two. And then one day trip. This is the specter of the composition of the Cuban metrics. I hope you understood. Thank you.

Good morning. Good morning or good afternoon. Uh, now we are supposed to find the spectral decomposition of the Matrix. So in order to find the spectral decomposition of a given matrix, we are supposed to first find the island values, uh, by using the characteristic equation, which is which is the given matrix minus Lambda Times? I question zero. So we need to make this determinant and then find us. I can values, uh, which let us suppose, uh, let us obtain three Eigen values. Let us just say that we obtain the Eigen values using this characteristic equation. And then we need to find the Eigen vectors corresponding to these respective values. Once we find out that I can like Eigen vectors of the corresponding I can values, what we can do is we can, uh, let assume that the human you do a new three vida. Uh, corresponding. I normalize. I convicted softer. I got victims, which we found that it takes one extra next three. So once we find out you, when you do a new thing, we can find a battery. Composition of the given metric Says equals two lander one times You want interview one Transport? Yes, Younger. Two times. You're good. I think you could transport Just land three times. Yeah, u three in two years. Three times. No, this is the way to find a spectral decomposition of the Cuban with Lambda one. Number two And number three are the three Eigen values that we obtained from the characteristic equation. And you, will you do your new three artist? Corresponding Normalize Dragon. Victors of the I can really use Lambda one number two and number three suspected me now You mean this one minus two six minus one? Yes. One minus one and fight. No, we can use the characteristic question and we can find the three either by use of this given metrics as Lambda equals to eight and brittle Question and number three Question three. No, we can find the now we can find them. Eigen vector sauce. Uh, given I can value said buy things that I did not Using the characteristic equation, you can find the Eigen vectors respectively. Now let X one b the Eigen vector of the Eigen values under one. It takes to be the victor of Eigen value. Let X tribute! I have a I have a new embassy. No. Excellent. Equals two minus one. Uh huh. Yeah. And then extra equal. Street minus one 12 and text three works through one. What? No, we need to again remember that we can find Eigen vectors by putting the again by using the characteristic equation which we obtain. No. Once we found out the Eigen vectors, like even u two and u three b the normalizing In matters of the Eigen vectors we found excellent. Next to an extreme perspective. Therefore, human becomes minus one one body. And you What we do is we divide, uh, given element of the organic automatics either, uh, square root of the sum of the squares of the elements. Dr. God is for me. Call them therefore for X one, which is minus 11 and zero, minus 11 and zero. What we do is we find out the normalized, uh, value by finding out minus world, which is an element divided by the square root of the summer squares of the column, which is minus one squared. Okay, minus one. Just a minute. Yeah. So it's a it's, like, route over part of my back. Yeah, just paper uh, minus one for this world. Yes, one calls First home sweat. Uh, my back and just a moment. So hold square. Yes. Zeros for and in the square root of the hold up which we find out this minus one day. So this is how we find an altar normalization similarly for all of the time, which is for X two, which is the cradle you two equals two minus 16 and then minus one. That'll change. And then remember, we can find the normalization with the U three where I can vector u c x three, which is impartial one barrel t one day one day. And then once we find notice Malaysia, Reagan, victors of the different Eigen vectors that we found out Then we can close the, uh can normalize I connectors Let that I can pick that might be in therefore and equals two. We are clubbing human usual man, you too. Don't we get matrix this And Okay, so we know we need not do this. What we can do is we can now, uh, find the security composition of the given matrix. Okay, which is equal to which I already told it, which is equal to equals. Two Lambda one. Thank you. Went into human transport issue number two times. You do it to transport three times three know what you can do is we can basically we can find her. Uh, for example, it is to find the value of you want to You want transport? So you want into You want transports? He goes to equals two, uh, minus one There, too. One bite of food. Zero. And then you want transport, which is? Yeah. No, we can use them to certification. And we would obtain a three by three matrix. It should be one by two. And this one guy zero and then minus one day, too. One by two, Vero. And then, you know, zero. Okay. Zero. So this is you want transport? Similarly, we can find the value of you to your transport and you 3 to 3 transport. And then finally, we can find a spectral decomposition of the Cuban made with a which is given like this, and the spectral decomposition would be equals two Lambda one which is equal to eight times you want to. Do you want France course, which is uh huh. Minuses. Zero and the Highness off. Uh huh. And zero. And then zero zero. Yeah, plus number two, which is equal to six times you think you're doing. So you could run, which is one by six on my six, minus two by six. And then one by six, one by six and then managed to basics and then minus two by six minus two by six. Okay, for about six, and then we can find a All the terms are won by three. One bite one by three, one by three, one by three, one bite. Uh, we can just find it. I'm just writing down the value of 23 times with three transports. This is the spectral decomposition of the given matrix. Thank you. I hope this is good. Thank you. Thanks a lot.

To find out a spectral decomposition of a given metrics. So in order to find the spectral decomposition of a given matrix, we need to find the Eigen values and then the Eigen vectors, and then the normalized Eigen vectors of the respective Eigen vectors. And then we can proceed further by, uh, the next steps. So basically, we can find the Eigen values of a given matrix by using the characteristic equation I've left. Given Matrix is a then a minus lambda I equals to zero. So we need to make this determinant, and then we can find out Eigen values. No, Let lander one minute or I beg your pardon given me tricks. Then let us suppose that x one x two and x three or the Eigen vectors of the corresponding I can value slam down number two and number three, respectively. Therefore x one x two and x three after Eigen vectors. After corresponding, I can values number one number two. Number three. Now, once we find out extra extra next three, we can normalize the Eigen vectors and then you can find the normalized Eigen vectors which are which letters. Suppose you won't be there normalize Eigen, Vector of Eigen. Vector X one let you to be that of x two and let you three be the normalized like a matter of X three. I cannot talk now, once we find you when you to a new three. Let me let us suppose that there be a metric species. Uh, no. Sorry. Now we can find a spectral decomposition of the metrics by using the equation. E equals two lambda, one times human into human transport plus number two times you, too into you to transport. Plus under three times you tree into you. Three transports. No lambda one number two and number three other Eigen values that we obtain by using the characteristic equations. No, let's start with the question. So the given value of the matrix A which is equal to six minus two minus one minus two six minus one minus one minus one. Right. So this is the metrics. No. We can use the characteristic equation and find that I can values this Number one equals to eight. The three equals two. Three. Just a minute. Yeah, I'm the three equals 23 So now we can move on and we can find Eigen victors by putting the Eigen values corresponding and the characteristic equation that we found out now let expire next to an excavator. I get rid of corresponding Eigen values slammed down number two and number three respectively. So we obtain X one as X one equals two minus 110 extra equals two minus one minus one and two and extra equals two 111 Now let you win. You win, You three be the normalized Convict us also, I can Victor 61 x two and X three, respectively. So now you have one equals two, You will send you one equals two minus one. Baruto, Uh, one by two and zero. So this is the normalized. I can Victor off. I can. Victor. Excellent. Now you two equal. So minus one minute, six minus one were up. Six. And to buy route six. This is the normalized. I can Victor off. I can exclude this pendant. You can find you three, which is the normalized. I can Victor of icann Victor. Now, once we found out the one U two and u three now we can find you one into you want transport now you want to You want transpose equals. So now you witness minus one Baruto, one by roto and zero. This is the Matrix, and then you can't transpose equals two minus one Baruto one by Roberto and zero. So this is the metrics. And then we can obtain a three by three matrix. By this, as this is the three by one matrix, and then this is one by three metrics. So therefore, we would obtain the resultant metrics as a three by three matrix, which is equal to the values are of find myself zero and then minus off half zero and then zero zero zero. Now, similarly, we can find you two into two transports and you three to you three times. And then, as we know, we can use this equation. This give any question this, uh, have any question? As, uh, now we can use this question as it to use it so that we can find out the spectral decomposition. No. Therefore the equals two. I'm the one times which is eight times off. You went into human transport, which is half minus half and zero and then minus half half zero. My inner self half and then zero and then zero zero. Uh huh. Zero Mhm. Sorry. Plus Lambda coupons. But we one by six. One by six minus two by six And then one by six. One by six and minus two by six and then minus two by six minus two by six. 48 6 please. Lambda three times, which is equal to three times off u three and two u three transports, which is equal to all the elements, are equal one by three. My treat My three 123 Cheap. So this is how we find our perspective. The composition of the Cuban matrix A. I hope you like it. Thanks a lot. Thank you.


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