Transcription of The Matrix Exponential - UMass Lowell
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The Matrix Exponential - UMass Lowell
The Matrix Exponential (with exercises)by Dan and comments are Matrix ExponentialFor eachn ncomplex matrixA, define theexponentialofAto be the Matrix (1)eA= k=0 Akk!=I+A+12!A2+13!A3+ It is not difficult to show that this sum converges for all complex matricesAof any finitedimension. But we will not prove this a 1 1 Matrix [t], theneA= [et], by the Maclaurin series formula for the functiony=et. More generally, ifDis a diagonal Matrix having diagonal entriesd1,d2, .. ,dn,then we haveeD=I+D+12!D2+ = 10 001 ..00 0 1 + d10 00d2 ..00 0dn + d212!0 00d212! ..00 0d212! + = ed10 00ed2 ..00 0edn The situation is more complicated for matrices that are not diagonal. However, if amatrixAhappens to bediagonalizable, there is a simple algorithm for computingeA, aconsequence of the following A and P be complex n n matrices, and suppose that P is invertible.
6 Linear Systems of Ordinary Differential Equations Suppose that y = f(x) is a differentiable function of a real (scalar) variable x, and that y0= ky, where k is a (scalar) constant.In calculus this differential equation is solved by
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