The state equation of a second order system is

$$x(t) = Ax(t),\,\,\,\,x(0)$$ is the initial condition.

Suppose $$\lambda_1$$ and $$\lambda_2$$ are two distinct eigenvalues of A and $$v_1$$ and $$v_2$$ are the corresponding eigenvectors. For constants $$\alpha_1$$ and $$\alpha_2$$, the solution, $$x(t)$$, of the state equation is

Let $$\alpha$$, $$\beta$$ two non-zero real numbers and v₁, v₂ be two non-zero real vectors of size 3 $$\times$$ 1. Suppose that v₁ and v₂ satisfy $$v_1^T{v_2} = 0$$, $$v_1^T{v_1} = 1$$ and $$v_2^T{v_2} = 1$$. Let A be the 3 $$\times$$ 3 matrix given by :

A = $$\alpha$$v₁$$v_1^T$$ + $$\beta$$v₂$$v_2^T$$

The eigen values of A are __________.

The rank of the matrix $$\left[ {\matrix{ 1 & { - 1} & 0 & 0 & 0 \cr 0 & 0 & 1 & { - 1} & 0 \cr 0 & 1 & { - 1} & 0 & 0 \cr { - 1} & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 1 & { - 1} \cr } } \right]$$ is __________.

If the vectors $${e_1} = \left( {1,0,2} \right),\,{e_2} = \left( {0,1,0} \right)$$ and $${e_3} = \left( { - 2,0,1} \right)$$ form an orthogonal basis of the three dimensional real space $${R^3},$$ then the vectors $$u = \left( {4,3, - 3} \right) \in {R^3}$$ can be expressed as