Find the Jordan canonical form J for the matrix A. You need not determine an invertible matrix S

Find the Jordan canonical form J for the matrix A. You need...

Key Concepts

Jordan Canonical Form

The Jordan canonical form is a block-diagonal matrix that represents a linear operator or a matrix in a form where each block (called a Jordan block) corresponds to an eigenvalue and its generalized eigenvectors. This form highlights the structure of the matrix and makes it easier to study its properties such as powers of the matrix, exponentials, or solving differential equations by exposing the algebraic and geometric multiplicities of the eigenvalues.

Eigenvalues

Eigenvalues are scalars associated with a matrix or linear transformation that, when subtracted from the main diagonal of the matrix, yield a singular matrix. They are fundamental in understanding the behavior of linear systems because they indicate the scaling factors in the directions given by their corresponding eigenvectors. The process of finding eigenvalues involves solving the characteristic polynomial of the matrix.

Algebraic Multiplicity

The algebraic multiplicity of an eigenvalue is the number of times that eigenvalue appears as a root in the characteristic polynomial of the matrix. This concept is essential because it determines the total size of the Jordan blocks associated with that eigenvalue in the Jordan canonical form.

Geometric Multiplicity

The geometric multiplicity of an eigenvalue is the number of linearly independent eigenvectors associated with that eigenvalue. It provides insight into the number and sizes of the Jordan blocks; specifically, it equals the number of Jordan blocks corresponding to that eigenvalue when the matrix is transformed into its Jordan canonical form.

Jordan Blocks

Jordan blocks are square submatrices in the Jordan canonical form that take on a nearly diagonal structure, with an eigenvalue on the main diagonal and ones on the superdiagonal, if any block has size greater than one. These blocks represent the coupling between eigenvectors and generalized eigenvectors and are key to understanding the effect of non-diagonalizable behavior of the matrix.

Generalized Eigenvectors

Generalized eigenvectors extend the concept of eigenvectors to cases where the matrix does not have a complete basis of eigenvectors. They satisfy a chain of equations involving powers of the matrix minus the eigenvalue times the identity matrix. These chains form the columns of the matrices that bring the original matrix into Jordan canonical form and are crucial for constructing the Jordan blocks when the geometric multiplicity is less than the algebraic multiplicity.

Transcript

00:01 Hey everyone.

00:01 So the problem we're looking at today gives us this matrix a, and it asks us to find the jordan canonical form.

00:08 And that's all we have to do.

00:10 And you'll see this is actually deceptively simple for a 5 by 5 matrix because it is upper triangular and the diagonal is all 0.

00:18 So what upper triangular means is that everything below the diagonal is zero and the diagonal also being zero just simplifies this particular problem even further.

00:26 And the reason we care that it's upper triangular is because the determinant of the upper triangular matrix, is just their diagonal elements multiplied together.

00:35 So this would be 0 to the 5th, because there are 5 zeros long the diagonal, right? so determinant of a would be 0.

00:42 But we don't actually care about the determinant of a.

00:44 We care about the eigenvalues.

00:47 So we care about setting debt of a minus lambda i, right, equal to zero and solving for lambda.

00:56 So that means that, well, the determinant of a minus lambda i is also upper triangular.

01:01 So it's lambda to the fifth is going to be the determinant of a minus lambda i.

01:08 And that's going to be equal to zero.

01:11 It's actually going to be negative, but you can just divide that out.

01:14 It doesn't really matter.

01:15 But let's keep that there because it's going to be negative landers, right? and so what we see is that we immediately get the eigenvalue, lambda equals zero, and the multiplicity is going to be equal to five.

01:28 So you know that the jordan canonical form is going to have zeros, all down the diagonal.

01:34 But we need to determine where to put the ones.

01:38 Okay, so basically what we need to know is how many blocks is this jordan canonical form going to have? how many jordan blocks? and we know that that's going to be equal to the number of linearly independent eigenvectors for a, which is equivalent to solving this, the standard eigenvector equation, a minus lambda i.

02:00 So that's just going to be a minus zero i.

02:04 So that's going to be a.

02:06 You multiply it by v, an unknown v, set that to zero.

02:11 So a v equals zero.

02:12 We need to find basically the for this particular problem, we need to find the dimension of the null space for a...

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