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In Exercises 19_24_ determine whether A is diagonalizable_ SO, find a matrix P that diagonalizes the matrix A, and determine P-IAP19. A = -3 =321. A =...

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In Exercises 19_24_ determine whether A is diagonalizable_ SO, find a matrix P that diagonalizes the matrix A, and determine P-IAP19. A = -3 =321. A =

In Exercises 19_24_ determine whether A is diagonalizable_ SO, find a matrix P that diagonalizes the matrix A, and determine P-IAP 19. A = -3 =3 21. A =



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Find the geometric and algebraic multiplicity of each eigenvalue of the matrix $A$, and determine whether $A$ is diagonalizable. If $A$ is diagonalizable, then find a matrix $P$ that diagonalizes $A,$ and find $P^{-1} A P.$ $$A=\left[\begin{array}{ccc}19 & -9 & -6 \\25 & -11 & -9 \\17 & -9 & -4\end{array}\right]$$

Hello there. Okay, so in this exercise we got this matrix A. And we need to uh first for all the values of these matrix, we need to calculate the algebraic and geometric multiplicity. Okay, so let's start in this case, the asian values is really easy to calculate because this is a lower triangular matrix and story here. I got them uh A copy from here. This here is one, so is a lower triangular matrix. That means that the Eigen values are located at the diagonal of these matrix. Okay, so it is clear that the Eigen values are going to be zero with algebraic multiplicity equals to two because appeared two times and λ two equals to one with algebraic Multiplicity equals to one. Great. Now we need to get the geometric multiplicity and for that we need to consider the associated the Eigen space for each of the Eigen values. That means that for example, for lambda, one equals to zero. We need to consider this system and find all the non trivial solutions. So that system is equivalent to this case is just a. X equals two. So we got this system Extracts three equals 2, And the solution of this is that we got to free variables X two and X three. So that means that X two is going to be equal to t. For example, an X three is going to be equal to us. And the Based on this equation here, we got that X one, it's equal two minus s over three. So this gives us the general solution of the of this system, so is equals two, T 010 plus s -1, 01. So what happened is that in this case the Eigen space associated to this Eigen value here is generated by two vectors. Is the span of these two vectors. So these two our Eigen vectors. So actually we can choose C equals to one. To obtain the first Eigen vector, that's going to be 010 and as equals to three. And the second Eigen vector it's going to be minus 103 Great. So for number one equals to zero, we got to Eigen vectors. One is equal to 1010 and the second one is equal two -103. So we got two linearly independent Eigen vectors are associated to love the one. That means that the geometric multiplicity in this case is also equal to two. And as we know from what we calculate before the age of like multiplicity is also constitute. So both multiplicity hours are the same. So we are in the in a good way we can probably, we can analyze actually we can and the second I can value is he close to one? So we must find one Eigen vector. So we repeat the procedure we considered the system and we find the non trivial solutions. In this case this system is equivalent to -1000 -10300 x one X two X three equals to the zero vector. Mhm. The solutions of this system, well this first were reduced to the echelon form And we obtain 1000 1000 X three. Yeah. And the solutions is that X one is the cost to zero X two is also a cost to zero and here X three because we don't have any pilot is a free variable, So X three is equal to some constant value. And that means that the general solution is generated by just one vector is the span of this vector, Which means that the geometric multiplicity is one. Okay, so the number of Eigen vectors associated to each Eigen value will give you the geometric multiplicity. So in this case for lambda too, the geometric multiplicity and the algebraic multiplicity are the same. Great. So that's the first part of this exercise. No. Based on this result, we have checked that for for all the Eigen values, the geometric multiplicity is equal to the anti black multiplicity. And by the theorem 5.2.4, it says that eight is diagonal. Izabal if and only if for all the Eigen values of a the geometric multiplicity is close to the edge of wreck multiplicity, which is this case. So we can organize these matrix and the the process is not that hard. So we need to find the the matrix speed and the matrix speed is generated by the it has in the columns, the corresponding Eigen vectors. Okay, so that means that in this case I'm going to great here again. The Eigen vectors or I N vectors are 010 -103. And the third that we just computed previously Is 001. So you can put in any other actually, but in this case I'm going to put in the other. They obtained them. So that means 010 -103 And 001. Right? So we got P. And then we need to calculate um to complete the organization, the members of the so the embers of B Is equal to 301 -100 and 01. Great. Um yeah, I'm going to to put this I'm going to shift this to because otherwise this doesn't make sense. Yes, here is 010 and 001 It doesn't matter how the order that you put the iron vectors on the metric speed. Okay, so here we got P and P inverse. So I'm going to copy them here again. So P is able to 001 -103 010 And the inverse of this matrix is 301 -100 and 010. And what happened is that we are going to use this to calculate the diagonal matrix. That is supposed to be obtained after a plane PM burst A P um after playing this where A is equal 2000000301 This diagonal matrix is equal to 00000001. Is that they have no metrics, as you can see. But the most important thing that they want you to notice here is that this, the econometrics is actually formed by the Eigen values of the matrix A. Then we competed before, and that's it. Thank you.

Okay, so for problem 22 were first given Matrix K, which is diagnosed Technol Izabal. So that means by definition we can have a have any murderball metrics say ass times as he first time says is De were where these are diving a matrix. Excuse me. Now also, by the definition off similarity then were given that be similar to a So we also have, um t inverse times. Hey, times t she's bi. Of course, T has to be a murderer matrix. Now we first soft a by multiplying I'm on applying t from the left hand side of right hand side so that mrs, uh, sorry. Second question. We must have a which is tee times be I'm stealing verse. Now we put the D. C. Based compression into our first question. No, we have asked members times t hey, t inverse times asked. Okay, Now here. Uh, yeah, I I missed the Dagenham matrix D on the right hand side. Now here, uh, let's take you to be a scene first times t so that, um our yeah, so our you immerse will be t t inverse times has. Okay, So since these two matrices are all in murderball, so you has to be in veritable so we can rewrite our expression here. D to be you times B times Ewing burst. So we've shown that be, by definition, bees.

So in order to see if this matrix a equals 13 minus 9 25 minus 17 is diagonal Izabal. We need to find that I can rallies of a But in order to do that, we need to find the determinant of a minus lambda times. The identity matrix said that equal to zero and sulfur lampa. So the determinant will be 13 minus lambda minus 9 25 minus 17 minus lander. So what set this equal to zero? So taking this determinant, we have 30 and minus lampa times minus 17 minus lambda minus negative. Nine times 25 equal standard. So multiplying this out, we have minus 2 21 minus 30 in Lambda Plus 17 Lambda plus Lambda Square plus 2 25 equals zero. So combining like terms, we have slammed a squared plus four lambda plus four equals zero. Now we factor this will get lambda plus two squared equals zero. Therefore, our Aiken value will be lambda equals negative to with the multiplicity of to now, in order to see if the matrix is diagonal sizable. What? We need to find a basis for the Eiken space for Lambda equals minus two and we can do that by taking the null space of the Matrix a minus minus two times the Identity Matron's, which is just no a plus to ah, so we need to find an all space of the matrix 13 plus two minus nine 25 minus 17 plus two. So this will be 15 minus nine 25 minus 50. Now we divide the top of I 15 We'll get one minus 3/5, 25 minus 15. And now we multiply the top row by 25 subtract this second row from it. We'll get one minus 3/5 00 Now we're called up the notes. Face of a matrix is the set of solutions to the equation. A X equals zero. So the north space of this matrix we'll just be exit to times the vector 3/5 want. Now we see that the Eiken space of the matrix has a dimension of what? Since there's only one factor, however, we found one I can value with the multiplicity of to That means that in order for the matrix to be diagonal Izabal, that Eigen space had to have a dimension of tube, which means that we needed to have two factors

Hello there. So for this exercise we got a general to biometrics and we need to in case that this diagram risible finding their criminalization of these general matrix. So it is important to record them result. That is the general formula for the Eigen values of these metrics. So the Eigen values for these matrix is given by these formula that they have correct in here. So you're concerned. We got two solutions that's given by this pleasant minus. And what is inside of the square root is called the discriminate. And I'm going to write in this way by this pre angle simple. So what happened is that if the discrimination is greater than zero, then we got to solutions to different solutions for the environment. That means that these metrics is that journalism? It gives it is equal to zero. Then we got to generate value. And for two x two matrices we cannot have the generate Eigen values because we can only have one Eigen vector for each of the values. And that implies that if this criminal is equal to zero, then this metric is not organized. So we're going to choose this case and find the Eigen vectors. So for that let me express for the first case, when lambda plus Is equal to 1/2 times a plus D plus minus the square root of this pre ankle, I may use made use of this to um make the calculations easier. So we need to find the solutions for this system and that is equal to have after. Well this is equal to one half. I'm going to put the whole process a minus d minus the square root of. So here is here it is the plus sign, two, BC. And here 1/2 the minus a minus the square root of the track. X one X two equals to the 00 factory. When we reduce this matrix deviation form, what we obtain is the following retain 1/2. Will we just reduce these metrics to eliminate the second world? And we obtain the following zero here. zero a dynasty minus the square root of triangle B. And X one X two Equals 20 victory. And what happened, I'm going to write here again. So we got λ plus that is equal to 1/2 A plus B plus the square root of triangle. And our system a minus lambda plus identity. Ex ecos zero vector is equivalent to one half a minus d minus the square root of triangle. B. 00 Okay. And the solution for the system is X one equals two minus B and X two equals to what is here? One half a minus d minus the square root of triangle. That means that the Eigen vector v plus Is equal to -7. One health a minus d minus the square root of prayer. And you can observe something that is that if we change to the other, I can value that is with the minus sign here. So lambda minus. The only thing that changed is design here and they sign here and therefore design here. So we obtained a plus B minus the school of um Delta is the name of this triangle. Yes, upper case. Uh delta. And this is equal two -71 health A minus D plus the square root of delta. So we got our two Eigen five years here so we can construct the metrics be so I'm going to look right here, we got lambda plus minus that is equal to one half a plus B plus minus the square root of delta in this or discriminate to be properly saying And those criminals equals to four BC minus a minus the square. And the corresponding again valley Eigen vectors of these two Eigen values R equals 2 -7 minus one. Health A minus d minus the square root of plus minus. We'll hear the relation change is minus plus of delta. Okay, so the metrics be at the end of the decriminalization of this matrix A. These are general matrix is equal to P. Matrix is equals two minus B minus B. One health A minus D or space minus this group of delta and minus B. One half of a dynasty plus the square root of health or the indiscriminate. And that this is the matrix speed that diagonal eyes A means A. Is written ap that diagonal matrix B in person.


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