Let the sets of eigenvalues and eigenvectors of a matrix B be $$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvectors of the matrix A, where $$B = {P^{ - 1}}AP$$, respectively, are

The value of the line integral $$\int_P^Q {({z^2}dx + 3{y^2}dy + 2xz\,dz)} $$ along the straight line joining the points $$P(1,1,2)$$ and $$Q(2,3,1)$$ is

Let $$x$$ be an $$n \times 1$$ real column vector with length $$l = \sqrt {{x^T}x} $$. The trace of the matrix $$P = x{x^T}$$ is

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