doubt on matrix exponential

Question

I have the following expression:

$e^{At}=e^{A_1t}u_1v^{T}_1 + e^{A_2t}u_2v^{T}_2$

where A is $A=\begin{pmatrix} 0 & 1\\ 1& 0 \end{pmatrix}$ and $u_1=\begin{pmatrix} 1\\ 1 \end{pmatrix}$ and $u_2=\begin{pmatrix} 1\\ -1 \end{pmatrix}$ are the eigenvectors associated to the eigenvalues $\lambda _1=1$ and $\lambda _2=-1$.

Now, what I don't understand is what is $v^{T}$.

I have also the expression :

$$T = \begin{pmatrix} v_1^{T}\\ v_1^{T} \end{pmatrix} =\begin{pmatrix} u_1 & u_2 \end{pmatrix}^{-1}$$

which I don't understand.

Can somebody please help me? Thanks in advance.

Answer 1

This is just diagonalization of A and then exponentiating. If A is diagonalizable, then

$A \begin{bmatrix}v_1 && v_2 \end{bmatrix} = \begin{bmatrix}v_1 && v_2 \end{bmatrix} \begin{bmatrix} \lambda_1 && 0 \\ 0 && \lambda_2 \end{bmatrix} $

where $\lambda_1$ and $\lambda_2$ are eigenvalues and $v_1$ and $v_2$ are corresponding eigenvectors. Then A can be written as:

$A = \begin{bmatrix}v_1 && v_2 \end{bmatrix} \begin{bmatrix} \lambda_1 && 0 \\ 0 && \lambda_2 \end{bmatrix} \begin{bmatrix}v_1 && v_2 \end{bmatrix}^{-1} $

Since in your example the eigenvectors are orthogonal, $\begin{bmatrix}v_1 && v_2 \end{bmatrix}^{-1} = \begin{bmatrix}v_1 && v_2 \end{bmatrix}^T $.

Now in Taylor expansion of $e^{At}$ we have sum of powers of $At$, for example

$A^2 = \begin{bmatrix}v_1 && v_2 \end{bmatrix} \begin{bmatrix} \lambda_1 && 0 \\ 0 && \lambda_2 \end{bmatrix} \begin{bmatrix}v_1 && v_2 \end{bmatrix}^{-1} \begin{bmatrix}v_1 && v_2 \end{bmatrix} \begin{bmatrix} \lambda_1 && 0 \\ 0 && \lambda_2 \end{bmatrix} \begin{bmatrix}v_1 && v_2 \end{bmatrix}^{-1} = \begin{bmatrix}v_1 && v_2 \end{bmatrix} \begin{bmatrix} \lambda_1 && 0 \\ 0 && \lambda_2 \end{bmatrix}^2 \begin{bmatrix}v_1 && v_2 \end{bmatrix}^{-1}$

And similarly,

$A^n = \begin{bmatrix}v_1 && v_2 \end{bmatrix} \begin{bmatrix} \lambda_1 && 0 \\ 0 && \lambda_2 \end{bmatrix}^n \begin{bmatrix}v_1 && v_2 \end{bmatrix}^{-1}$

And therefore,

$e^{At} =\begin{bmatrix}v_1 && v_2 \end{bmatrix} e^{\begin{bmatrix} \lambda_1 && 0 \\ 0 && \lambda_2 \end{bmatrix}t} \begin{bmatrix}v_1 && v_2 \end{bmatrix}^{-1} = \begin{bmatrix}v_1 && v_2 \end{bmatrix} \begin{bmatrix} e^{\lambda_1 t} && 0 \\ 0 && e^{\lambda_2 t} \end{bmatrix} \begin{bmatrix}v_1 && v_2 \end{bmatrix}^{-1} = \begin{bmatrix}v_1 && v_2 \end{bmatrix} \begin{bmatrix} e^{\lambda_1 t} && 0 \\ 0 && e^{\lambda_2 t} \end{bmatrix} \begin{bmatrix}v_1 && v_2 \end{bmatrix}^{T} $