Let T be an incomplete matrix, and suppose 𝐰1, …, 𝐰j is a Jordan chain associated with an incomp

Let T be an incomplete matrix, and suppose 𝐰1, …, 𝐰j is a...

Key Concepts

Jordan Chains

A Jordan chain is a sequence of generalized eigenvectors associated with an eigenvalue of a matrix, especially when the matrix is not diagonalizable. These chains extend the concept of eigenvectors to cases where the geometric multiplicity is less than the algebraic multiplicity, systematically constructing vectors that satisfy a hierarchy of equations involving (T - ?I). This framework is fundamental in understanding the structure of matrices that lead to Jordan blocks.

Generalized Eigenvectors

Generalized eigenvectors are vectors that satisfy (T - ?I)^k v = 0 for some positive integer k, rather than the standard eigenvalue equation (T - ?I) v = 0. They fill in the missing pieces when a matrix does not have a complete set of linearly independent eigenvectors. They are essential in constructing Jordan chains and, consequently, the Jordan canonical form.

Jordan Basis

A Jordan basis is a complete set of vectors consisting of eigenvectors and generalized eigenvectors such that the matrix representation becomes as close as possible to a diagonal form; specifically, it becomes block diagonal with Jordan blocks corresponding to each eigenvalue. This basis simplifies understanding the action of the matrix, particularly when computing functions of the matrix, like powers used in iterative processes.

Matrix Exponentiation and Binomial Expansion

When a matrix is expressed in its Jordan form, raising it to a power involves a modified binomial expansion. In particular, the action of T^k on a generalized eigenvector (or a vector in a Jordan chain) can be described by a series similar to the binomial expansion, where powers of the eigenvalue are combined with combinatorial coefficients and lower chain vectors. This encapsulates the idea that the non-diagonal entries contribute additional terms proportional to k, k choose 2, etc.

Iterative System Solution using Jordan Form

For a linear iterative system of the form u^(k+1) = T u^(k), transforming T into its Jordan canonical form allows one to decouple the system into simpler components. By using the Jordan basis, the evolution of the system can be expressed explicitly in terms of the eigenvalues and the structure of the Jordan blocks, resulting in a general solution that consists of linear combinations of terms involving powers of eigenvalues and polynomial factors reflecting the block sizes.

Transcript

00:01 Let's start this exercise by remembering that the transpose of a diagonal matrix is itself.

00:13 So the diagonal lambda 1, lambda 2, lambda n is equal to the transpose of itself.

00:30 And if we use our hypothesis, which is that this diagonal matrix can be written as s inverse times a times s then this transpose would be s inverse times a times s transposed which is the same as s transpose times a transpose times s inverse transpose now we can write s transpose as s transpose inverse inverse inverse and then times a transpose times s inverse transposed.

01:13 Now i'll set in the exercise if we let q to the s inverse transpose then when we write this in terms of q we obtain q inverse a transpose times q and this is going to be equal to the diagonal matrix lambda 1, 2, lambda n.

01:42 And this proves the first part of our exercise.

01:48 For the second part, it is important to remember that s inverse is 1 over the determinant of s times the transpose of the matrix of cofactors.

02:06 And we can also write this as the determinant of s times s inverse transpose is equal to the matrix of cofactors.

02:27 So as we said before, d is equal to its transposed.

02:34 And again, as we did in part a, d transpose is the same as s inverse times a times s transpose, which is s transpose times a transpose times s inverse transposed, and we know that s inverse here, if we use our first relation, we know that s inverse is the same as one over the determinant times the transpose of the matrix of k factors, and we're going to put that the 1 over the determinant of s at the beginning, so it's a scalar, times s transpose times a transposed times mc transposed, which is the same as 1 over the determinant of s, a transpose times the matrix of cofactors.

03:53 Now from here, if we rewrite this equation, so we take d equals to this part, we have, we put the determinant of s on the other side times z equals s transpose times a transpose, and it's the matrix of a factors...

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