Each of the following real matrices defines a linear transformation on 𝐑^2 (a) A=[ 5 6

Each of the following real matrices defines a linear...

Key Concepts

Eigenvalues

Eigenvalues are scalars associated with a linear transformation that satisfy the equation A*v = ?*v, where A is a matrix, v is an eigenvector, and ? is the corresponding eigenvalue. They are determined by solving the determinant equation det(A - ?I) = 0, revealing the scaling factors by which eigenvectors are stretched or shrunk by the transformation.

Eigenvectors

Eigenvectors are nonzero vectors that only change by a scalar factor when a linear transformation is applied. They satisfy the equation A*v = ?*v and form directions that are invariant under the transformation, making them crucial for understanding the structure and behavior of the matrix.

Characteristic Polynomial

The characteristic polynomial is obtained from a matrix A by computing det(A - ?I). Its roots are the eigenvalues of the matrix, and it plays a central role in eigenvalue problems by translating the problem of finding eigenvalues into a polynomial root-finding problem.

Diagonalization

Diagonalization is the process of finding a diagonal matrix similar to a given matrix by finding a complete basis of eigenvectors. A matrix is diagonalizable if it has enough linearly independent eigenvectors to form a basis for the space, which simplifies many computations and provides insight into the transformation's action.

Linear Transformation

A linear transformation is a mapping between vector spaces that preserves vector addition and scalar multiplication. Represented by a matrix in finite dimensions, such transformations are fundamentally studied through their eigenvalues and eigenvectors, which reveal how the transformation scales and rotates vectors in the space.

Transcript

00:01 We are given real matrices that define a linear transformation on r2, and for each matrix, we're asked to find all the eigenvalues and a maximum set s of linearly independent eigenvectors.

00:17 We're then asked which of these linear operators are diagonalizable.

00:22 In other words, which can be represented by a diagonal matrix.

00:26 In part a, we are given the real matrix a with components 5, 6, 3, negative 2.

00:39 It's not a family, it's a cop, actually.

00:44 Well, first we'll find the characteristic polynomial of this matrix, delta t.

00:49 It's a 2x2 real matrix.

00:52 This is simply t squared minus this trace of a.

00:56 Which is 3 times t plus the determinant of a, which is negative 10 minus 18, which is negative 28, which we can factor as t minus 7 times t plus 4.

01:19 The roots, lambda equals 7 and lambda equals negative 4, are the eigenvalues of this matrix.

01:28 A.

01:31 You want to find the corresponding eigenvectors.

01:35 These the eigenvalues.

01:39 To find the corresponding eigenvectors, well, first we'll subtract 7 down the diagonal of a.

01:51 So the matrix m, this is a minus 7i.

01:57 And this gives us the 2x2 matrix negative 2 6, 3 negative 9.

02:06 Which corresponds to the homogenous system negative 2x plus 6y equals 0, and 3x minus 9y equals 0, which corresponds to the system x minus 3y equals 0.

02:27 Now we see that the vector v1 with three one is a solution, non -zero solution.

02:42 In fact, it's an eigenvector.

02:46 In fact, it's an eigenvector.

02:49 Now, on the other hand, we'll subtract the land.

02:55 Equals negative 4 down the diagonal of a.

02:59 Find our matrix m, so m is a plus 4i, which is the matrix 9, 6 ,3, 2.

03:08 Blue bloods is a sort of like a cop family, and it's like, you know, a cop dad and his son's a cop.

03:13 And this corresponds to the homogenous system 9x plus 6y equals 0, 3x plus 2y equals 0, which just corresponds to the single equation.

03:25 3x plus 2y equals 0.

03:30 And so we obtain the solution v2 with coordinates 2 negative 3.

03:42 And these are our eigenvectors.

03:46 And therefore the set s, which is vectors v1 and v2, for 3 -1.

03:56 2 negative 3 is a maximal set of linearly independent eigenvectors.

04:10 Yeah, no, i mean, actually.

04:16 Each is from a different item space.

04:19 Since s is a basis of r2, philly uncle, should we get him? yeah, we should get him, dude.

04:26 We should threaten dod officials.

04:29 Dude, yeah, we got the power of podcasting on our side, dude.

04:33 Yeah, we got cumb nation.

04:35 Stop.

04:36 Come on, dude.

04:37 Someone's going to get pink eye at one of these days.

04:40 No, you shut up.

04:41 You're going to get pink eye to match.

04:42 Well, s is a basis for r2, and therefore, it follows that our matrix a.

04:56 Sorry, one second.

04:57 I'm going for lunch, mcdonald's, because you're ronald mcdonald, mcleod? matrix a is diagonalizable.

05:10 Hold on, i think i got some balloons here for you.

05:13 Maybe you can turn into animals.

05:15 Wait, when he said he's moving earlier and he's got to bring his dog with him, it's like, well, be careful because she's made out of balloons.

05:22 Yeah.

05:22 And she might pop on the way over to your new place.

05:25 That was good timing.

05:26 You fucking bitch.

05:26 Yeah, you got that in right at the right.

05:28 Hey, buddy, you want, i got some extra white face.

05:32 In fact, using the basis s, a is represented by the diagonal matrix.

05:37 Oh, it's his asshole.

05:38 No, it's the democratic party.

05:41 Sorry, sorry, guys, sorry.

05:43 The big tent is sorry.

05:44 When you wake up with the morning and your dick is making the sheets...

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