MIT18.06 Linear Algebra (22)

lec22

We have $n$ independent eigenvectors of $A$, and put them in columns of matrix $S$.

$AS$

$$AS=\begin{bmatrix} \vdots & & \vdots\\ \lambda_1x_1 & \cdots & \lambda_nx_n\\ \vdots & & \vdots \end{bmatrix}= \begin{bmatrix} \vdots & & \vdots\\ x_1 & \cdots & x_n\\ \vdots & & \vdots \end{bmatrix} \begin{bmatrix} \lambda_1 & 0 & 0\\ 0 & \ddots & 0\\ 0 & 0 & \lambda_n \end{bmatrix} =S\Lambda$$

if $n$ eigen vectors are independent,

$$\begin{aligned} AS & = S\Lambda \\ S^{-1}AS & = \Lambda\\ A & = S\Lambda S^{-1} \end{aligned}$$

eigenvalues of $A^2$

if

$$Ax=\lambda x$$

$$A^2x=\lambda Ax=\lambda^2 x$$

also

$$A^2 = S\Lambda S^{-1}S\Lambda S^{-1}=S\Lambda^2S^{-1}$$

$$A^k = S\Lambda^kS^{-1}$$

$A $has **independent eigenvectors** if all$ \lambda$s are different.

If have repeated eigenvalues, may or may not have n independent eigenvectors.

$u$

start with given vector $u_0$

$$⁍$$

$$\rightarrow u_k=A^k u_0$$

solve $u_{100}$

write $u_0$ as combination of eigenvectors.

$$u_0 = c_1x_1+c_2x_2+\cdots+c_nx_n$$

then calculate $Au_0$:

$$Au_0 = c_1Ax_1+c_2Ax_2+\cdots+c_nAx_n \\ = c_1 \lambda_1 x_1+c_2\lambda_2 x_2+\cdots+c_n\lambda_n x_n \\$$

$$A^{100}u_0 = c_1 \lambda_1^{100} x_1+c_2\lambda_2^{100} x_2+\cdots+c_n\lambda_n^{100} x_n \\$$

$$A u_{0}=S \Lambda S^{-1} u_{0}=S \Lambda S^{-1}(S c)=S \Lambda c\\ u_{100} = A^{100} u_{0}=S \Lambda^{100} c$$

Example of Fib. Sequence

$$F_{k+2}=F_{k+1}+F_{k}$$

$$u_k = \left[\begin{array}{c}F_{k+1} \\F_{k}\end{array}\right],u_{k+1}=\left[\begin{array}{ll}1 & 1 \\1 & 0\end{array}\right]\left[\begin{array}{c}F_{k+1} \\F_{k}\end{array}\right]=A u_k$$

Find eigenvalue of $A$:

$$\lambda ^2 - \lambda -1=0 \Rightarrow \lambda =\frac{1\pm \sqrt[]{5} }{2}$$