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Introduction Examples Proving why Fulmer’s Method works for A k Example of e At from A k Summary Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker July 12, 2013 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Proving why Fulmer’s Method works for A k Example of e At from A k Summary Table of Contents Introduction 1 Why? Definitions Examples 2 Partial Fractions Decomposition Fulmer’s Method Proving why Fulmer’s Method works for A k 3 Linear Independence Generalizing to an n × n matrix Example of e At from A k 4 Summary 5 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Why find A k and e At ? A k is essential to find the solutions to difference equations. Calculating e At , the matrix exponential. e At is used in solving matrix linear differential equations. Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Definition Let n be a nonnegative integer. The falling factorial is the sequence k n , with k = 0 , 1 , 2 , . . . given by the following formula. Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Definition Let n be a nonnegative integer. The falling factorial is the sequence k n , with k = 0 , 1 , 2 , . . . given by the following formula. k n = k ( k − 1)( k − 2) · · · ( k − n + 1) . If k were allowed to be a real variable then k n could be characterized as the unique monic polynomial of degree n that vanishes at 0 , 1 , . . . , n − 1. Observe also that k n | k = n = n !. Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Definition Let n be a nonnegative integer and a be a complex number. We define a sequence ϕ n , a ( k ) as � a k − n k n a � = 0 , n ! ϕ n , a ( k ) = δ n ( k ) a = 0 . Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Definition Let n be a nonnegative integer and a be a complex number. We define a sequence ϕ n , a ( k ) as � a k − n k n a � = 0 , n ! ϕ n , a ( k ) = δ n ( k ) a = 0 . where � 0 k � = n δ n ( k ) = 1 k = n Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Example With ϕ 0 , 0 ( k ) , ϕ 1 , 4 ( k ) , and ϕ 2 , 5 ( k ) , we have Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Example With ϕ 0 , 0 ( k ) , ϕ 1 , 4 ( k ) , and ϕ 2 , 5 ( k ) , we have ϕ 0 , 0 ( k ) = δ 0 ( k ) = (1 , 0 , 0 , 0 , 0 , . . . ) Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Example With ϕ 0 , 0 ( k ) , ϕ 1 , 4 ( k ) , and ϕ 2 , 5 ( k ) , we have ϕ 0 , 0 ( k ) = δ 0 ( k ) = (1 , 0 , 0 , 0 , 0 , . . . ) = 4 k − 1 k ϕ 1 , 4 ( k ) = (0 , 1 , 8 , 48 , 256 , . . . ) Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Example With ϕ 0 , 0 ( k ) , ϕ 1 , 4 ( k ) , and ϕ 2 , 5 ( k ) , we have ϕ 0 , 0 ( k ) = δ 0 ( k ) = (1 , 0 , 0 , 0 , 0 , . . . ) = 4 k − 1 k ϕ 1 , 4 ( k ) = (0 , 1 , 8 , 48 , 256 , . . . ) = 5 k − 2 k ( k − 1) ϕ 2 , 5 ( k ) = (0 , 0 , 1 , 15 , 150 , . . . ) 2 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Definition Let y ( k ) be a sequence of complex numbers. We define the Z -transform of y ( k ) to be the function Z{ y ( k ) } ( z ), where z is a complex variable, by the following formula: ∞ y ( k ) � Z{ y ( k ) } ( z ) = z k k =0 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Now that we’ve defined the Z -Transform, we can now apply it to ϕ n , a . Let a ∈ C and n ∈ N and we obtain the following formula. z Z{ ϕ n , a ( k ) } ( z ) = ( z − a ) n +1 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Theorem Let A be an n x n matrix with entries in the complex plane. Then Z{ A k } ( z ) = z ( zI − A ) − 1 where I is the n x n identity matrix. Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Why? Proving why Fulmer’s Method works for A k Definitions Example of e At from A k Summary Note that the Z -Transform is one-to-one and linear. Therefore, the Z -Transform has an inverse. Now that we know that the Z -Transform is invertable we obtain the following formula A k = Z − 1 { z ( zI − A ) − 1 } Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Partial Fractions Decomposition Proving why Fulmer’s Method works for A k Fulmer’s Method Example of e At from A k Summary Example Find A k if � 2 − 1 � A = 1 0 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Partial Fractions Decomposition Proving why Fulmer’s Method works for A k Fulmer’s Method Example of e At from A k Summary Recall Z{ A k } ( z ) = z ( zI − A ) − 1 . First, we would compute zI − A . � z − 2 � 1 zI − A = − 1 z Next, compute the inverse of zI − A . 1 � z − 1 � ( zI − A ) − 1 = z 2 − 2 z + 1 1 z − 2 1 � z − 1 � = ( z − 1) 2 1 z − 2 � − 1 � z ( z − 1) 2 ( z − 1) 2 = 1 z − 2 ( z − 1) 2 ( z − 1) 2 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Partial Fractions Decomposition Proving why Fulmer’s Method works for A k Fulmer’s Method Example of e At from A k Summary So we must perform partial fraction decomposition to obtain z A B ( z − 1) 2 = ( z − 1) + ( z − 1) 2 z = A ( z − 1) + B If z = 1, 1 = A (1 − 1) + B 1 = A (0) + B 1 = 0 + B B = 1 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Partial Fractions Decomposition Proving why Fulmer’s Method works for A k Fulmer’s Method Example of e At from A k Summary z A B ( z − 1) 2 = ( z − 1) + ( z − 1) 2 z = A ( z − 1) + B If z = 0, 0 = A (0 − 1) + B 0 = A ( − 1) + B 0 = − A + B A = B A = 1 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Partial Fractions Decomposition Proving why Fulmer’s Method works for A k Fulmer’s Method Example of e At from A k Summary Plug in results: z 1 1 ( z − 1) 2 = ( z − 1) + ( z − 1) 2 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Partial Fractions Decomposition Proving why Fulmer’s Method works for A k Fulmer’s Method Example of e At from A k Summary z − 2 A B ( z − 1) 2 = ( z − 1) + ( z − 1) 2 z − 2 = A ( z − 1) + B If z = 1, 1 − 2 = A (1 − 1) + B − 1 = A (0) + B − 1 = 0 + B B = − 1 Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker

Introduction Examples Partial Fractions Decomposition Proving why Fulmer’s Method works for A k Fulmer’s Method Example of e At from A k Summary z − 2 A B ( z − 1) 2 = ( z − 1) + ( z − 1) 2 z − 2 = A ( z − 1) + B If z = 0, 0 − 2 = A (0 − 1) + B − 2 = A ( − 1) + B − 2 = − A + B 2 + B = A 2 + ( − 1) = A 1 = A Calculating A k using Fulmer’s Method Rasheen Alexander, Katie Huston, Thomas Le, Camera Whicker