Question

Explain why a discrete time system is stable if its eigenvalues are all in the unit circle.

Explain why a symmetric matrix will always have real eigenvalues.

Explain why a continuous time system is stable if the real part of its eigenvalues are negative.

Answer 1

1) for discrete systems, the eigenvalues must lie inside the unit circle, that is, their modulus must be less than one.

as, a discrete time system is asymptotically stable if and only if all eigenvalues of its state matrix are inside the unit circle .

2) let A be a symmetric matrix with eigenvalue

combined with gives

now,

as the quotient of a non-negative real number by a positive one

so, must be real

3) We know that the general solution is x(t) = eAtx0

thus, x(t) 0 if and only if as

we will now show that this happens if and only if all the eigenvalues of A have negative real parts

let

be the jordan canonical form for A

then,

let be the eigenvalue of a associated with Ji

then e Jit will tend to 0

if and only if has negative real parts

therefore, tend to 0 if and only i all the eigenvalues of A have negative real parts