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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 5 39 Copyright © Orchard Publications Summary The Cayley Hamilton theorem states that a matrix can be expressed as an degree polynomial in terms of the matrix as where the coefficients are functions of the eigenvalues . If all eigenvalues of a given matrix are distinct, that is, if the coefficients are found from the simultaneous solution of the system of equations If some or all eigenvalues of matrix are repeated, that is, if the coefficients of the state transition matrix are found from the simultaneous solution of the system of equations n1 () th A e At a 0 Ia 1 Aa 2 A 2 a A ++ + + = a i λ A λ 1 λ 2 λ 3 …λ n ≠≠≠≠ a i a 0 a 1 λ 1 a 2 λ 1 2 a λ 1 +++ + e λ 1 t = a 0 a 1 λ 2 a 2 λ 2 2 a λ 2 + e λ 2 t = a 0 a 1 λ n a 2 λ n 2 a λ n + e λ n t = A λ 1 λ 2 = λ 3 = m , λ m1 + λ n = a i a 0 a 1 λ 1 a 2 λ 1 2 a λ 1 + e λ 1 t = d d λ 1 -------- a 0 a 1 λ 1 a 2 λ 1 2 a λ 1 + d d λ 1 --------e λ 1 t = d 2 d λ 1 2 0 a 1 λ 1 a 2 λ 1 2 a λ 1 + d 2 d λ 1 2 λ 1 t = d d λ 1 --------------- a 0 a 1 λ 1 a 2 λ 1 2 a λ 1 + d d λ 1 ---------------e λ 1 t = a 0 a 1 λ + a 2 λ + 2 a λ + + e λ + t = a 0 a 1 λ n a 2 λ n 2 a λ n + e λ n t =

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Chapter 5 State Variables and State Equations 5 40 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications We can use the MATLAB eig(x) function to find the eigenvalues of an matrix. A column vector that satisfies the relation where is an matrix and is a scalar number, is called an eigenvector. There is a different eigenvector for each eigenvalue. Eigenvectors are generally expressed as unit eigenvectors, that is, they are normalized to unit length. This is done by dividing each component of the eigenvector by the square root of the sum of the squares of their components, so that the sum of the squares of their components is equal to unity.
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