Cyclic groups
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The Cayley-Hamilton theorem
Suppose that a 2 x 2 matrix A has characteristic equation [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.][Equation goes here - download the original to see it.] In this argument we have made use of the fact that [Equation goes here - download the original to see it.] The Cayley-Hamilton theorem can be established for square matrices of any dimension. This theorem is used as a computational device for finding inverses of matrices and higher powers. Example You are given the matrix [Equation goes here - download the original to see it.] (a) Show that 0 and 1 are eigenvalues of [symbol], and find the other eigenvalue. (b) Using the Cayley Hamilton theorem, or otherwise (i) Show that [Equation goes here - download the original to see it.] Verify that the matrix M satisfies its own characteristic equation (iii) Explain why it is not possible to obtain an value for [Equation goes here - download the original to see it.] by cancelling down and rearranging the characteristic equation. Solution (a) The characteristic equation is [Equation goes here - download the original to see it.] If [symbol] we obtain [Equation goes here - download the original to see it.] If [symbol] we obtain [Equation goes here - download the original to see it.] If [Equation goes here - download the original to see it.], we observe that the above equation holds. Therefore, [Equation goes here - download the original to see it.] are eigenvalues. The characteristic equation yields[Equation goes here - download the original to see it.] (b)(i) The characteristic equation is [Equation goes here - download the original to see it.] Therefore, from Cayley-Hamilton theorem, we have [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] (ii) The characteristic equation is [Equation goes here - download the original to see it.] and we have [Equation goes here - download the original to see it.] To verify this equation we compute [Equation goes here - download the original to see it.] (iii) To obtain an inverse we would like to argue as follows [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.]
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Contents of
Cyclic groups
1 Cyclic Groups
2 The Cayley-Hamilton theorem
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