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Unformatted text preview: Matrix Exponentials Math 246, Fall 2009, Professor David Levermore We now consider the homogeneous constant coefficient, vector-valued initial-value problem (1) d x d t = Ax , x ( t I ) = x I , where A is a constant n × n real matrix. A special fundamental matrix associated with this problem is the solution Φ ( t ) of the matrix-valued initial-value problem (2) d Φ d t = AΦ , Φ (0) = I , where I is the n × n identity matrix. We assert that Φ ( t ) satisfies (i) Φ ( t + s ) = Φ ( t ) Φ ( s ) for every t and s in R , (ii) Φ ( t ) Φ ( − t ) = I for every t in R . Assertion (i) follows because both sides satisfy the matrix-valued initial-value problem d Ψ d t = AΨ , Ψ (0) = Φ ( s ) , and are therefore equal. Assertion (ii) follows by setting s = − t in assertion (i) and using the fact Φ (0) = I . The fundamental matrix Φ ( t ) is therefore called the exponential of A and is commonly denoted as either e t A or exp( t A ). It is easy to check that the solution of the initial-value problem (1) is given by x ( t ) = e ( t − t I ) A x I . The Taylor expansion of e t A about t = 0 is (3) e t A = ∞ summationdisplay k =0 1 k ! t k A k = I + t A + 1 2 t 2 A 2 + 1 6 t 3 A 3 + 1 24 t 4 A 4 + ··· , where we define A = I . Recall that the Taylor expansion of e at is e at = ∞ summationdisplay k =0 1 k ! a k t k = 1 + at + 1 2 a 2 t 2 + 1 6 a 3 t 3 + 1 24 a 4 t 4 + ··· . Motivated by this fact, the book defines e t A by the infinite series (3). Matrix KEY Identity. Given any polynomial p ( z ) = π z m + π 1 z m − 1 + ··· + π m − 1 z + π m and any n × n matrix A we define the n × n matrix p ( A ) by p ( A ) = π A m + π 1 A m − 1 + ··· + π m − 1 A + π m I . Because for every nonnegative integer k one has d k d t k e t A = A k e t A , it follows from the definition of p ( A ) that (4) p parenleftbigg d d t parenrightbigg e t A = p ( A ) e t A . This is the matrix version of the KEY identity. Just as the scalar KEY identity allowed us to construct explicit solutions to higher-order linear differential equations with constant coefficients, the matrix KEY identity allows us to construct explicit solutions to first-order linear differential systems with a constant coefficient matrix. 1 2 Computing the Matrix Exponential. Given any n × n matrix A , there are many ways to compute e A t that are easier than evaluating the infinite series (3). The book gives a method that is based on computing the eigenvectors and (sometimes) the generalized eigenvectors of the matrix A . This method requires a different approach depending on whether the eigenvalues of the real matrix A are real, complex conjugate, or have multiplicity greater than one. These approaches are covered in Sections 7.5, 7.6, and 7.8, but these sections do not cover all the possible cases that can arise. Here we will give a different method that covers all possible cases with a single approach. Moreover, this method is generally much faster to carry out than the book’s method when...
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