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Eigen values give you the characteristic equation of a system, in fact when you solve a an RLC circuit using differential equations Aλ^2+Bλ+C=D(λ), the solution to Aλ^2+Bλ+C =0, are the eigen values of the system.. Therefore the geometrical significance of eigen values is that is gives you the position of the poles of the system.
From a geometrical point of view, if we look at the states of each variable of the system to be controlled, every "bad" state of these variables must be controllable and observable to ensure a good behaviour in the closed-loop system. That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the close-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis.
Geometrically, a Matrix A can see like a transformation axes.
The Matrix A acting about a determined axes, don't produce rotation, but only stretching, this special axes, are the eigen vector, this special factor of streching are the eigenvalues.
Let V be a column vector ( 3 X 1). Each element in the column vector can be thought of as the componet of a "VECTOR" in three mutually perpendicular axes.
And A be a (3X3) matrix.
Now if i compute A*V we will get another column vector (3 x 1) say U
i.e. A*V = U
These V and U may or may not have same orientation or direction. They may or maynot have same magnitude. So because of this multiplication A*V, i am getting another vector U, which is a stretched (or compressed) and direction changed version of V.
If both U and V have the same orientation, then this multiplication A*V is equivalent to multiplication by a scalar (λ) as far as V is concerned, as the direction is unchanged. Then this particular vector V is called EIGEN VECTOR of matrix A. So for the EIGEN VECTOR V, A*V =λV. This scalar λ corresponding to EIGEN VECTOR V is called the EIGEN VALUE.
So for the EIGEN VECTOR V, multiplication by the matrix A can be replaced by multiplication by the scalar λ.
We can note that for each EIGEN VALUE there will be a number of EIGEN VECTORS, each one related to the other by a scale factor. Because of that EIGEN VECTORS are computed for unit magnitude.
EIGEN VECTORS and EIGEN VALUES are derived from the Matrix involved.
EIGEN VALUES have some properties
1. Sum of the EIGEN VALUES is equal to the sum of the elements in the main diagonal of A.
2.Product of EIGEN VALUES is equal to the Determinant A.
Read book by Lari Moore on Adaptive signal processing.
The eigen value of a matrix is defined as follows : if Ax = λx then λ is called the eigen value of the matrix A.
In continuous time signals: e^*(j*Wo*t) is an eigen value for a LSI system H,
since
H(x) = λ*x ---- where x = e^(j*Wo*t)
i.e If the i/p to an LSI system is a sinusoid, then the output is scaled
(possibly time shifted) version of that sinusoid.
When a signal can be represented as a sum of sinusoids using fourier
transform/series, i.e as a sum of eigen values. The output of a LSI system can
be represented as weighted sum of the eigen values/ sinusoids. And the weights
are given by the transfer function of LSI system "H".
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