Question

When computing the eigenvalue decomposition of a matrix 'A', what is the nature of the solution vector 'x(t)' obtained using the matrix exponential?

A) Linear in 't'
B) Exponential in 't'
C) Periodic in 't'
D) Constant in 't'

Answer 1

The solution vector 'x(t)' obtained using the matrix exponential is exponential in 't'.

Step-by-step explanation:

The solution vector 'x(t)' obtained using the matrix exponential when computing the eigenvalue decomposition of a matrix 'A' is exponential in 't'.

The matrix exponential is defined as e^(At), where 'e' is the base of the natural logarithm and 't' is the time variable. The exponential nature of the solution vector 'x(t)' means that it grows or decays exponentially with time.

For example, if 'A' has eigenvalues with positive real parts, the solution vector 'x(t)' will grow exponentially as 't' increases. On the other hand, if 'A' has eigenvalues with negative real parts, the solution vector 'x(t)' will decay exponentially as 't' increases.