Which of the matrices in the set S in Problem 14 have a set of exactly five (and not more than f

Which of the matrices in the set S in Problem 14 have a set...

Key Concepts

Algebraic and Geometric Multiplicity

The algebraic multiplicity of an eigenvalue is its number of repetitions as a root of the characteristic polynomial, while the geometric multiplicity is the number of independent eigenvectors associated with that eigenvalue. For a matrix to have a full set of eigenvectors—specifically, exactly as many as the dimension of the space (such as five independent eigenvectors in a 5×5 matrix)—the geometric multiplicity must equal the algebraic multiplicity for each eigenvalue.

Jordan Canonical Form

The Jordan canonical form restructures a matrix into blocks (Jordan blocks) that highlight the structure of its eigenvalues, particularly when the matrix is not fully diagonalizable. This form helps in understanding situations where there are fewer independent eigenvectors than the size of the matrix, emphasizing the relationship between the algebraic and geometric multiplicities of eigenvalues.

Diagonalizability

Diagonalizability is a property of matrices that have a complete set of linearly independent eigenvectors. A diagonalizable matrix can be expressed in the form PDP?¹, where D is a diagonal matrix of eigenvalues and P is the matrix of corresponding eigenvectors. This property is important because it dramatically simplifies many operations, like computing matrix powers or exponentials.

Eigenvectors

Eigenvectors are the nonzero vectors that remain parallel after a linear transformation is applied, changing only in scale (not direction). The collection of eigenvectors, when linearly independent, can form a basis for the vector space, which is crucial for understanding and simplifying the behavior of matrices via diagonalization.

Eigenvalues

Eigenvalues are the scalars associated with a linear transformation that satisfy the equation A*v = ?*v for some nonzero vector v. They indicate the factors by which the eigenvectors are scaled and are central in determining the behavior of the transformation, including stretching or compressing along certain directions.

Transcript

0:00 Hey everyone.

00:01 So the problem we're looking at today builds off of a previous problem.

00:05 I did a video on that too.

00:06 This is the final screen we got.

00:08 If you guys watch that video, if you didn't, feel free to check it out.

00:11 Or if you're fine on that problem, and you need to help with this.

00:14 Just a little tour of what we have.

00:18 We have right here.

00:20 We have written our eigenvectors and their multiplicity.

00:24 These are all the different permutations, additive permutations, we can get to equal their multiplicity.

00:31 So there's seven, for the fives, two for the negative, seven for the sixes, two for the negative twos.

00:37 And that's just equivalent to the different number of ways we can break up each set of eigenvalues here in our jordan canonical form into different jordan blocks.

00:47 Okay.

00:48 So without further ado, the new question.

00:52 Basically, it's asking us, well, not basically, it is asking us which, which matrices in a set we wrote down, which was, if you remember, the set of all, possible jordan canonical forms for something with this characteristic polynomial.

01:10 So which of those matrices have exactly five linearly independent eigenvectors? now, this is a question about a certain theorem that you have in your textbook, which states that the number of jordan blocks is equal to the number of linearly independent eigenvectors for the matrix.

01:31 So basically rewarded, the question asks us for which matrices in the set we wrote down earlier.

01:37 So which possible jordan canonical forms have exactly five jordan canonical.

01:43 Jordan blocks, sorry, jordan blocks in their jordan canonical form.

01:47 So we can go through our list of possibilities right here for each eigenvalue, right? and we can make combinations that add up to five.

02:00 And all of those are going to be answers to this question.

02:07 All of those matrices are going to have five and exactly five linearly independent eye connectors.

02:13 So the numbers here, remember, are the sizes of the blocks.

02:18 So this is one block of size two, and the amount of things added.

02:23 So this is one block of size one and one block of size one.

02:26 So this is actually two blocks here.

02:31 Yeah, that's two blocks there.

02:33 And over here is going to be one so don't just add two, like you could say that, i don't know, two plus three is five, but this is actually one block, so you need one, and then in the next one you need four blocks.

02:48 Right, so just don't get confused there.

02:50 It's the number of blocks, not the size of the blocks that we're looking at.

02:53 So if we have one block in the twos, and remember that's going to be a block of size two, so just one right there.

03:02 We need how many different ways can we get four blocks in, the sixes...

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