Eigenvalue
, Eigenvector Analysis of a Square Matrix A
Example 2
.
![A = matrix([[3, 0, -5], [1/5, -1, 0], [1, 1, -2]]), ` `*(A-I*lambda) = matrix([[3-lambda, 0, -5], [1/5, -1-lambda, 0], [1, 1, -2-lambda]])](images/EigenvaluesExample22.gif)
![evecsAvec1 = vector([5, 1, 3])](images/EigenvaluesExample26.gif){kind=small}
![A-I*lambda = matrix([[3, 0, -5], [1/5, -1, 0], [1, 1, -2]])](images/EigenvaluesExample28.gif)
![matrix([[3, 0, -5], [1/5, -1, 0], [1, 1, -2]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/EigenvaluesExample29.gif)
![matrix([[1, 0, -5/3], [0, 1, -1/3], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/EigenvaluesExample210.gif)
![matrix([[a-5/3*c], [b-1/3*c], [0]]) = matrix([[0], [0], [0]])](images/EigenvaluesExample211.gif)
![v[1] = matrix([[5], [1], [3]])](images/EigenvaluesExample212.gif)
![evecsAvec2 = vector([5*2^(1/2)+5, 1, 1+2*2^(1/2)])](images/EigenvaluesExample215.gif){kind=small}
![A-I*lambda = matrix([[3-2^(1/2), 0, -5], [1/5, -1-2^(1/2), 0], [1, 1, -2-2^(1/2)]])](images/EigenvaluesExample217.gif)
![matrix([[3-2^(1/2), 0, -5], [1/5, -1-2^(1/2), 0], [1, 1, -2-2^(1/2)]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/EigenvaluesExample218.gif)
![matrix([[1, 0, 5/(-3+2^(1/2))], [0, 1, (-1+2^(1/2))/(-3+2^(1/2))], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/EigenvaluesExample219.gif)
![matrix([[a+5/(-3+2^(1/2))*c], [b+(-1+2^(1/2))/(-3+2^(1/2))*c], [0]]) = matrix([[0], [0], [0]])](images/EigenvaluesExample220.gif)
![v[2] = matrix([[5*2^(1/2)+5], [1], [1+2*2^(1/2)]])](images/EigenvaluesExample221.gif)
![evecsAvec3 = vector([-5*2^(1/2)+5, 1, 1-2*2^(1/2)])](images/EigenvaluesExample224.gif){kind=small}
![A-I*lambda = matrix([[3+2^(1/2), 0, -5], [1/5, -1+2^(1/2), 0], [1, 1, -2+2^(1/2)]])](images/EigenvaluesExample226.gif)
![matrix([[3+2^(1/2), 0, -5], [1/5, -1+2^(1/2), 0], [1, 1, -2+2^(1/2)]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/EigenvaluesExample227.gif)
![matrix([[1, 0, -5/(3+2^(1/2))], [0, 1, (1+2^(1/2))/(3+2^(1/2))], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/EigenvaluesExample228.gif)
![matrix([[a-5/(3+2^(1/2))*c], [b+(1+2^(1/2))/(3+2^(1/2))*c], [0]]) = matrix([[0], [0], [0]])](images/EigenvaluesExample229.gif)
![v[3] = matrix([[-5*2^(1/2)+5], [1], [1-2*2^(1/2)]])](images/EigenvaluesExample230.gif)
![v[1] = matrix([[5], [1], [3]]), v[2] = matrix([[5*2^(1/2)+5], [1], [1+2*2^(1/2)]]), v[3] = matrix([[-5*2^(1/2)+5], [1], [1-2*2^(1/2)]])](images/EigenvaluesExample232.gif)
![P = matrix([[5, 5*2^(1/2)+5, -5*2^(1/2)+5], [1, 1, 1], [3, 1+2*2^(1/2), 1-2*2^(1/2)]]), P^`-1` = matrix([[-1/5, 1/2, 1/2], [1/20*(1+2^(1/2))*2^(1/2), 1/8*(-2+2^(1/2))*2^(1/2), -1/4], [1/20*(-1+2^(1/2))...](images/EigenvaluesExample233.gif)
![P^`-1`*P*` = `*matrix([[-1/5, 1/2, 1/2], [1/20*(1+2^(1/2))*2^(1/2), 1/8*(-2+2^(1/2))*2^(1/2), -1/4], [1/20*(-1+2^(1/2))*2^(1/2), 1/8*(2+2^(1/2))*2^(1/2), -1/4]])*matrix([[5, 5*2^(1/2)+5, -5*2^(1/2)+5],...](images/EigenvaluesExample234.gif)
![P^`-1`*A*P*` = `*matrix([[-1/5, 1/2, 1/2], [1/20*(1+2^(1/2))*2^(1/2), 1/8*(-2+2^(1/2))*2^(1/2), -1/4], [1/20*(-1+2^(1/2))*2^(1/2), 1/8*(2+2^(1/2))*2^(1/2), -1/4]])*matrix([[3, 0, -5], [1/5, -1, 0], [1,...](images/EigenvaluesExample235.gif)