35 fundamental and exponential matrices

35 fundamental and exponential matrices - 18.03 Lecture#35...

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18.03 Lecture #35 Dec. 2, 2009: notes The topic for today is using (linear) algebra to say interesting things about solving systems of ODE. Most of what I’ll say today applies to systems that are linear and homogeneous : x ( t ) = A ( t ) x ( t ) (system of homog. linear ODE) with A ( t ) an n × n matrix of functions of t . But the most important case is A ( t ) = A , an n × n constant matrix, and that’s all that I’ll talk about: x ( t ) = A x ( t ) (system ODE) From time to time we’ll be interested in initial conditions x ( t 0 ) = x 0 . (Initial conditions) The fundamental algebraic fact about solutions to (system ODE) is a version of what I wrote at the end of the notes for Lecture 33: Superposition Theorem. Suppose x 1 , x 2 , . . . x m are solutions of (system ODE), and that c 1 , c 2 , . . . c m are real numbers. Then the linear combination x = c 1 x 1 + c 2 x 2 + · · · + c m x m is also a solution of (system ODE). What this says is that taking linear combinations of column vectors is important for solving systems of differential equations. The big idea today is that matrix multiplication takes linear combinations of column vectors . We will therefore be able to use matrix multiplication to help solve systems. Here’s a more precise statement. Suppose X is an n × m matrix. This means that X consists of m column vectors x 1 = x 11 x 21 . . . x n 1 , x 2 = x 12 x 22 . . . x n 2 , · · · x m = x 1 m x 2 m . . . x nm . If we apply X to the m × 1 column vector c = c 1 c 2 . . . c m , we get X · c = c 1 x 1 + c 2 x 2 + · · · + c m x m . That is, multiplying a matrix X times a column vector c forms a linear combination of the columns of X . 1
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Here’s a generalization. Suppose C is an m × p matrix, with p columns c 1 = c 11 c 21 . . . c m 1 , c 2 = c 12 c 22 . . . c m 2 , · · · c p = c 1 p c 2 p . . . c mp . Then the matrix product XC is n × p : it has p columns each of height n . The columns are X · c 1 , X · c 2 , · · · X · c p . That is, the th column of XC is a linear combination of the columns of X , with coefficients given by the th column of C . Example. 1 0 1 1 0 1 parenleftbigg 3 5 2 6 parenrightbigg = 3 5 5 11 2 6 . The first column of the product is three times the first column, plus two times the second column. The second column of the product is five times the first column, plus six times the second column. Here’s a translation into ODE language. Superposition Corollary. Suppose X ( t ) is an n × m matrix-valued function of t , and that each column is a solution of (system ODE). That is, assume that X ( t ) satisfies the matrix ODE X ( t ) = AX ( t ) ( matrix system ODE ) (a system of n · m linear ODE, one for each entry of X ). Suppose also that C is an m × p constant matrix. Then the n × p matrix-valued function Y ( t ) = X ( t ) · C is a solution of (matrix system ODE).
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