Exponential - Exponential of 2 2 Matrices These notes...

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Exponential of 2 × 2 Matrices These notes describe how to calculate e tA for a 2 × 2 matrix A . We recall that by definition the exponential of a matrix is e tA = X n =0 t n A n n ! . For computations the basic relation we will use is e tP - 1 AP = P - 1 e tA P e tA = Pe tP - 1 AP P - 1 . There are three cases to consider: 1. A has real eigenvalues and is diagonalizable. 2. A has real eigenvalues but is not diagonalizable. 3. A has a pair of complex conjugate eigenvalues. Case 1: real eigenvalues & diagonalizable This case comes down to calculating the exponential of a diagonal matrix. Let B be the matrix B = λ 1 t 0 0 λ 2 t . It is easy to verify that B n = λ n 1 t n 0 0 λ n 2 t n . Therefore e B = X n =0 B n n ! = n =0 λ n 1 t n n ! 0 0 n =0 λ n 2 t n n ! = e λ 1 t 0 0 e λ 2 t . Returning to A , let λ 1 and λ 2 be its eigenvalues (note that λ 1 and λ 2 need not be distinct). In this case we can find two independent eigenvectors v 1 and v 2 corresponding to
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