Transcription of The Matrix Exponential - University of Massachusetts Lowell
{{id}} {{{paragraph}}}
Example: stock market
The Matrix Exponential - University of Massachusetts 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. TheneP 1AP=P that, for all integersm 0, we have(P 1AP)m=P 1 AmP.
3! A3 + It is not difficult to show that this sum converges for all complex matrices A of any finite dimension. But we will not prove this here. If A is a 1 t1 matrix [t], then eA = [e ], by the Maclaurin series formula for the function y = et. More generally, if D is a diagonal matrix having diagonal entries d 1,d 2,. . .,dn, then we have eD ...
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: