(a) Give an example of a matrix A such that A^2 has an eigenvector that is not an eigenvector of

(a) Give an example of a matrix A such that A^2 has an...

Key Concepts

Eigenvalues

Eigenvalues are scalars associated with a matrix that provide key information about its action on vectors. When a matrix acts on one of its eigenvectors, the output is simply the scalar (the eigenvalue) times that eigenvector. They play a central role in determining the behavior of matrix transformations, stability analysis, and many other properties in linear algebra.

Eigenvectors

Eigenvectors are nonzero vectors that only change by a scalar factor when a linear transformation (represented by a matrix) is applied. They are intimately connected to eigenvalues, as each eigenvalue corresponds to one or more eigenvectors. While the relationship is straightforward for a matrix, the eigenvectors of a matrix raised to a power do not necessarily coincide with the original matrix’s eigenvectors unless the matrix is well?behaved (for example, diagonalizable).

Matrix Powers

When a matrix is raised to a power, such as when computing A², the eigenvalues of the resulting matrix are related to those of the original matrix by the corresponding power (for instance, if ? is an eigenvalue of A, then ?² is an eigenvalue of A²). However, the eigenvectors of A² may include additional vectors beyond those that are eigenvectors of A, especially in cases where A is not diagonalizable. This phenomenon is important in understanding how matrix functions alter the spectral properties of matrices.

Spectral Mapping Theorem

The spectral mapping theorem establishes that when a matrix is transformed by a function – such as squaring – the eigenvalues of the new matrix correspond to the function of the eigenvalues of the original matrix. Specifically, if f(?) is the function applied (in this case, f(?) = ?²), then every eigenvalue of the squared matrix is the square of an eigenvalue of the original matrix. This theorem provides a theoretical foundation for understanding the relationship between the spectra of A and A².

Transcript

0:00 Hey everyone.

00:01 So today we're looking at a problem that asks us just to give an example of a 2x2 matrix, a.

00:09 So let's just write out a generic a, b, c, d.

00:18 It's bad d.

00:20 Anyways, and we want to get this matrix to have a generalized eigenvector that is not a regular eigenvector.

00:27 And in other words, we do not want this matrix to have two linearly independent eigenvectors.

00:37 So we want it to be defective because defective matrices are the only type of matrices that will have jordan canonical forms that are not diagonal.

00:47 And thus will have generalized eigenvectors as opposed to having only regular eigenvectors.

00:53 So how we're going to do that? honestly you can just play around with this, but if you're watching this video, it probably means that you would prefer a little nudge in the correct direction.

01:06 I certainly did need to take a little bit of a formulaic approach.

01:15 So if we think about our criteria that it needs to be defective, we can say that we need, well, first of all, there can't be two different eigenvalues because we can't be.

01:29 We can't be two different eigenvalues because we can't.

01:30 We know that different eigenvalues have linearly independent eigenvectors.

01:35 So the first thing we need to find is some way to make sure that a only has one eigenvector, and it'll have a multiplicity of two.

01:47 So the way we do that, so the way we find eigenvectors is we take a minus lambda i, and that equals a minus lambda b, c, d minus lambda, b, c, d minus lambda, right? okay.

02:11 And we need to take the determinant of that.

02:14 So debt a minus lambda i, debt that, right, is going to be, and we take this times this minus this, right? so that's going to be a minus lambda, d minus c, b.

02:40 And to find the eigenvalue.

02:41 And to find the eigenvalue, lambda, we need to set this, the determinant needs to be zero.

02:49 So we need to have this all equals zero, right? now, if we expand this out, we will get a quadratic equation and sparing you some algebra.

03:04 You get that lambda squared plus negative d minus a, lambda plus a d minus c b that's all going to equal zero that's what we want right so this is just a regular quadratic equation you can check that it comes from this that we know how to solve so what are the solutions to this using the quadratic formula x equals negative b which is going to be d minus a so d plus a plus or minus the square root of b squared.

03:52 So the square root of negative 1 is, i mean this, sorry, b squared, so b is negative 1 times d plus a, so can just write d plus a squared.

04:03 D plus a squared minus 4 ac minus 4a is 1.

04:12 So a, d, d minus c, so a, d minus c, be all over two.

04:20 Okay, so that's our equation.

04:22 Now, when will this have only one value? well, it gets its two values from this plus or minus.

04:29 So we need the plus to be the same as the minus.

04:32 The only number that is true for is zero.

04:35 So really our criteria boils down to this term right here in the square root being zero.

04:40 So we can rewrite this as d plus a squared minus 4 times a, d minus c, b equals zero.

04:57 And this is our equation that we need to fulfill in order to have a repeated eigenvalue.

05:05 So anything that fulfills this equation will have one eigenvalue and it will be repeated.

05:09 That being said, that doesn't necessarily mean that it is defective.

05:15 If it has one eigenvalue, it could have one eigenvalue with two eigenvectors, right? but there's no real point to try and go into the super nitty -gritty and like objectively, empirically solve for the specific exact formula that will always give you a defective matrix.

05:37 So basically this should just be a guideline for you.

05:41 It was for me and then i guessed some numbers that fit that.

05:45 So for example, i guessed 0, 1, negative 1, 2, where again i is defined like that.

05:58 And it turns out if we check the eigenvectors or sorry, eigenvalues of this, we do get a repeated number.

06:07 We get lambda equals 1 with a multiplicity of 2.

06:12 And then if we check the eigenvectors, that we check the eigenvectors, that we check the again vectors will you have to solve 0 minus 1, 1, negative 1, 2 minus 1.

06:25 Solve whatever vector will make that equal 0 if you multiply it so you can do an augmented matrix 0 0 that goes to 2 minus 1 1 minus 1 0...

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