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LetA = [-2 2 point) Set up the matrix Apoints) Find the det(A Al ) and identify the characteristic polynomial of A.points) List the eigenvalues of _ along with the ...

Question

LetA = [-2 2 point) Set up the matrix Apoints) Find the det(A Al ) and identify the characteristic polynomial of A.points) List the eigenvalues of _ along with the algebraie multiplicity of ech eigenvalue:(4 points) Find basis for the eigenspace corresponding t0 the eigenvalue 0. (Only find basis for this eigenspace You do nOt needto do this for the Other cigenvalue(s) )point) State the geomctric multiplicity of the cigenvalue

LetA = [-2 2 point) Set up the matrix A points) Find the det(A Al ) and identify the characteristic polynomial of A. points) List the eigenvalues of _ along with the algebraie multiplicity of ech eigenvalue: (4 points) Find basis for the eigenspace corresponding t0 the eigenvalue 0. (Only find basis for this eigenspace You do nOt needto do this for the Other cigenvalue(s) ) point) State the geomctric multiplicity of the cigenvalue



Answers

Find the characteristic equation, the eigenvalues, and bases for the eigenspaces of the matrix. $$\left[\begin{array}{rrr}1 & 0 & -2 \\0 & 0 & 0 \\-2 & 0 & 4\end{array}\right]$$

The topic of this question is icon values and Eigen vectors. The question asks us to find a basis for each Eigen space of the matrix and thus do all the steps leading up to that, including finding solving the characteristic equation and finding the organ rulings. So what is the characteristic equation? This equation is C. Of lambda equals zero where Sea of lambda is the characteristic polynomial defined by the determinant of mm minus lambda. I where um is what I am calling our matrix and I is the identity matrix. Yeah. So I want to write out and the minus lambda I and then take its determinant. So the determinant is a co factor expansion along any row or column of the matrix. So I'm going to choose this column since it has two zeros and that allows me to, that that gives me that I only have to compute the the factor of one entry. Every man is lambda. Yeah, since this entry is one of these five entries of this matrix, who's some of row and column numbers is even the multiplier of negative one to the power of something will be negative one to the power of an even number. And so multiplier is just one and I'm I'm multiplying this onto the sub matrix that I get or the determinant rather of the sub matrix I get by deleting Nero and column over the century. And that is this determinant. All the oh these entries would have a multiplier of negative one since their son of row and column numbers. It's hard now evaluating this two by two determinant, which is a d minus bc. We can evaluate our complete determinant. Sorry, this is not equal here. Since eight plus one, it's nine. Our complete factory ization is lambda minus three cubed. Since this is lambda minus three square. So we only have one Eigen value. Right, one solution to this characteristic equation. So we want to find a basis for the Eigen space corresponding because Eigen value. So now we know that this matrix has one, I can value of three. Now we want to find Eigen vectors. So since our lambda is three the matrix the un investable matrix we are interested in is lambda minus 35 Or this matrix. No this is a matrix of interest because it's no space is the Eigen space against the organ space of this. Eigen value is defined as the no space of the matrix m minus the psycho value turns the identity matrix. Now the easiest way I know how to um solve this system is just real, reduce the matrix the completely reduced form, it takes only one ro ro transformation to achieve and it is this so we're reducing doesn't change um the null space of the matrix. So the new space in this matrix will still be the null space on this one. And it's very easy to find the null space of this one. I won't write out all the zeros just for the sake of time, this tells us that X equals that. And so X, y z can be written as a linear combination of two vectors one oh one and 010 if you're not sure where this comes from, I got it from writing X, Y. Z as this linear combination, and I also use the fact that X equals said. So all solutions are all Eigen vectors can be written as linear combination of these two. And so these to form a basis for the Eigen space, there's only one Eigen value. So this is the only again, space we need has this basis

This question is about Eigen values and Eigen vectors here we want to find the Eigen values of this matrix and for each again, really find a basis for the ANC. In space. Recall that to find the Eigen values, we have to solve the characteristic equation which is characteristic polynomial equal zero. What is the characteristic polynomial? Again? It is in this case a cubic polynomial of the scalar lambda. Mhm. Defined by the determinant of a man assigned to. I worked as the identity nature. Yeah, this determinant is the determinant of this matrix here, which is a menace lambda I or just hey with negative land was added onto the middle or the deck. And or entry. Now I'm going to use the co factor expansion definition of the determinant define the three by three determinant here, one minus lambda. I'm going to choose this column. Take the co factor expansion of the column, which only has a non zero only has one non zero entry. The one corresponding to one climbed up and that will be one minus lambda times the co factor of what might have slammed down which is a positive one times the determinant of the sub matrix we get by deleting the column and row of this entry. So that's the co factor expansion of the middle room. Middle column one. Medicine. Bad times. It's co factor. Now we have to evaluate this two by two determinant, which is just a demon sbc. So here I've expanded the foreman assigned attempts. One man is named after and once we factor that completely, we end up with Yeah, our three organ values. One, two and right. So our matrix A. Has these three Eigen values 12 and three. Now we want to find Eigen vectors. So an Eigen vector corresponding to the first dragon volume is a solution of this system. The system is happened homogeneous. So I must have at least one solution, which is the zero solution trivial solution in other words. But since we said that since the lambda equals one was a solution to the characteristic equation. A menace, I has a determinant of zero and thus must be uninhabitable. And so there is not just one solution in this case, but infinitely many. And in fact we go about the process of finding Eigen values like this specifically so that we would have infinitely many solutions and not just the zero vector. We want non zero vectors, non zero vector solutions to this. And the only way to have that is this where Mhm. By the way, Eigen vectors already find that's being non zero. So how do we find the solutions? Well, one way is to will reduce those matrix to roll reduced echelon form. This is my real reduced echelon form which uh well from here to here, I haven't written down my steps but all I did was subtract this row times three from this room and then switch the rose. So this tells me not. Ah the no space of this matrix is the same as the null space of this matrix. And so the solution I want here is the same from the solution set of this equation is the same as the solution set of this equation. And so uh if I parameter rise my solutions I would find that all the solutions can be written as a scalar multiple of this factor 010 X. And Z must necessarily be zero. And why is my pretty variable? I have just 13 variable. There's only going to be one vector in the uh basis of the organ space because all vectors can be recognized This killer multiple of just one. A linear combination of one victory. Now let's move on to the second african space. The one corresponding to the second item value. This again space is the set of all vectors that solve this system. What is that now? We have the second organ Valium. Thanks to find all these this infinite number of solutions, we will reduce this matrix that I like to call the Eigen matrix of this Eigen value. Since we already call we have already all these times for with agonize the prefix why not call this an Eigen matrix Now I will reduce that to find a solution of the homogeneous system corresponding to this matrix. After five or six rural reductions. I would get this rail reduced echelon form. It was solution set is the set of all scalar multiples hug this vector. And so this vector is a basis factor for the dragon space. It forms a set all on its own. That is a basis. The second idea. No, The last. The Eigen matrix is a minus three I. Which is this. So a possible solution vector is minus 111 and again we have one free variable. And so the basis has only one vector, which is an inspector. There is our third basis. Yeah. And as a fun fact, these three vectors could be used to diagonal. Is this matrix, but that's a topic for another video.

11. We say that the determined off a determine off a minus Lumumba unit in years even to deal So tu minus from the bigger and zero zero through minus Lunda 000 Still, minors learned that with the Equus 20 from here we can say that Monday's equipped in tow to end too for the Agnes based E So for ah longer is equal to two a minus. London one is equal air martyr. Buying by better be is equal to zero. This ah gives Ah zero Matics is not the blind by me. Want me to be three is equal to zero So on the director is equal to Lunda point 10 and zero plus as you one and zero plus T or tell these are constants As you zero and one We're a boy s in town for tea. Belongs to the are with the rational numbers So we can say that they will be which is 10 and zero and 01 and zero and 00 And why and for and the pace since for the again Ah, space he so that I mentioned my going to space e ive three. That's why we can say that a is ah non defective

Problem number 13. Ah, we have at the determines off a minus Islam. The one equal to you so ah to minus Lunda two and two to running Minor thrown about three. You get the run in at the one negative one line. Islam, that is equal to zero. We get that. How long? The minus two known squared is equal to zero. So London is equipped to do you. And there we are the Eigen values. So who were Lambda is equal to two on. And as we know that a minus London for the blind by the unity or B Victor is equal to Dio. Please give that for, uh me one who are with anyone. Ah minus you mean three equals zero and still be to minus v. Perry is equal to zero. From here we can say that the the victory is equal to three, two and four. Uh which worms when their paces. Ah, for the agony space E one So that I mentioned of the again in space. Ah, anyone is equal to one. For how long? A is equal to zero on and a Linus lumber time The any team that defector is equal to zero this year that to doubly one minus there be three is equal to zero and ah, definitely two is equal to zero. So So Ah, on every vector 10 and to warms the bases on the agonist base e to so that I mentioned offer the agony space it is equal to also one has not equal to two that Roy, we can say that a is the effective.


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