Antigenic evolution In Exercise 8.7 .27 we modeled antigenic...

Antigenic evolution In Exercise 8.7 .27 we modeled antigenic evolution with the recursion $\mathbf{x}_{t+1}=A \mathbf{x}_{t},$ where $$A=\left[ \begin{array}{cc}{2} & {3} \\ {0} & {\frac{9}{10}}\end{array}\right]$$ (a) Using the initial condition $\mathbf{x}_{0}=\left[ \begin{array}{l}{1} \\ {1}\end{array}\right],$ express the solution to the recursion in terms of the eigenvectors and eigenvalues of $A .$ (b) Describe the long-term behavior of $\mathbf{x}_{t}$

Antigenic evolution In Exercise 8.7 .27 we modeled antigenic evolution with the recursion $\mathbf{x}_{t+1}=A \mathbf{x}_{t},$ where $$A=\left[ \begin{array}{cc}{2} & {3} \\ {0} & {\frac{9}{10}}\end{array}\right]$$
(a) Using the initial condition $\mathbf{x}_{0}=\left[ \begin{array}{l}{1} \\ {1}\end{array}\right],$ express the solution to the recursion in terms of the eigenvectors and eigenvalues of $A .$
(b) Describe the long-term behavior of $\mathbf{x}_{t}$

Key Concepts

Eigenvalues

Eigenvalues are the scalars associated with a matrix transformation that indicate how the eigenvectors are scaled during the transformation. In the context of recursions and dynamical systems, they are crucial because the magnitude of the eigenvalues determines the rate of growth or decay in each eigendirection, which in turn influences the long-term behavior of the system.

Eigenvectors

Eigenvectors are the nonzero vectors that only change by a scalar factor when a linear transformation is applied. They provide a basis in which the action of the transformation (or recurrence) can be decoupled into independent modes, making it possible to express the solution of the system in a simplified and insightful form.

Diagonalization

Diagonalization is the process of decomposing a matrix into a product involving a diagonal matrix of eigenvalues and a matrix of corresponding eigenvectors. This technique simplifies the calculation of matrix powers in recurrences, allowing each mode to be analyzed separately and revealing the underlying dynamics of the system.

Recurrence Relations

Recurrence relations describe the evolution of a system over discrete time steps where the state at each step is given by a linear function of the state at the previous step. They provide a framework for modeling and analyzing processes such as antigenic evolution by capturing how the current state influences future trajectories.

Stability Analysis

Stability analysis in linear systems involves examining the eigenvalues of the transformation matrix to determine whether the system's state converges to an equilibrium, diverges, or exhibits oscillatory behavior over time. The eigenvalue with the greatest absolute value (the dominant eigenvalue) often dictates the long-term behavior of the system.

Transcript

00:13 We want to solve the recursion nt plus 1 equals a nt, where a equals 2 ,30, 9 tenths, and n1, i'm sorry, n0 equals 1 -1.

00:37 The solution looks like nt equals p d to the t, p inverse, n0, where d is a diagonal, such that a equals p d p inverse.

00:53 Since d is a diagonal matrix that's similar to a, it's going to have the eigenvalues of a on its diagonal.

00:59 So let's start by finding the eigenvalues of a by taking the determinant of a minus lambda on the diagonal or lambda times the identity.

01:13 We get 2 minus lambda times 9 tenths minus lambda.

01:21 And the eigenvalues are the roots of this determined.

01:25 So we get lambda equals 2 or 9 tenths.

01:32 So we now have d, d equals 2, 9 tenths.

01:40 And we also have d to the t because exponentiating a diagonal matrix just involves exponentiating the diagonal entries.

01:51 So now to find p, we know that the columns of p are the eigenvectors of a so let's start by finding the eigenvalue which has i the eigenvector which has eigenvalue 2 so let's call that eigenvector x y my matrix multiplication we get 2x plus 3y and 9 tenths y though on the other hand because it's an eigenvector with eigenvalue 2 we have 2x 2y so 2x plus 3y equals 2x by looking at the top entries which means that 3y equals 0 and thus y equals 0 and the second equation well the second equation also gets us y equals 0 which means that y is 0 and x can be anything so let's pick x to be 1 the simplest thing we can't pick 0 because 0 0 is never an eyeing back so we have the first column of p.

03:07 Now to find the second column of p we do the same thing except instead of multiplying everything by two we multiply everything by nine -tenths and now we have the second equation is nine -tenths y equals nine -tenths why which doesn't tell it well so far why can be anything and then the second equation is 2x plus three y equals nine -tenths x and so if we y can be anything.

03:40 Let's figure out what x is in terms of y.

03:43 So we have x 2 minus 9 tenths equals minus 3y.

03:51 2 minus 9 tenths is 11 tenths.

03:55 So we have x equals minus 3y over 11 tenths, but divide by 11 tenths...

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