Question is from subject- Linear System Theory.Need help on part (b) and part (c) of the problem. part (a) is already solved. Problem 1 Matrix differentiation -Let A(t) be an n x n matriz, which dependson t, i.e. its elements are functions of t:(A(t)lij = aij(t). for i,j =1,...,nThe derivative A(t) of A(t) with respect to t, is also an n x n matriz and is defined by takingelement-wise derivatives:LÅ(t)lj = åj(t). for i, j = 1,...,nFor n xn differentiable matrices A1(t), Ag(t) prove:a)b)Using induction, prove that for n x n differentiable matrices A1(t), A2(t), ...ALE), we have:...1c)The erponential of a n x n matrix A is defined as the follouring:еxpAi!i-0where A°= I. Suppose A(t) and À(t) commute. Prove:dexp A(t) = Å(t) exp A(t)

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Description: Question is from subject- Linear System Theory.Need help on part (b) and part (c) of the problem. part (a) is already solved. Problem 1 Matrix differentiation -Let A(t) be an n x n matriz, which dependson t, i.e. its elements are functions of t:(A(t)

We have A(t) , n×n matrix depends on t. such that A(t)ij=aij(t), for i, j=1,2,...,n and element-wise…

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Problem 1 Matrix differentiation - . Let A(t) be an nxn matrir, which depends on t, i.c. its elements are functions of t: (A(0)lij = aij(t), for i, j = 1,...,n The derivative A(t) of A(t) with respect to t, is also an n xn matriz and is defined by taking element-uise derivatives: (Ä(t)ly = åij(t), for i, j = 1,...,n a) For n xn differentiable matrices A1(t), Ag(t) prove: (41(1) A2(t) = Å¡(t)Aa(t) + A1(t)Å½{t) Using induction, prove that for n x n differentiable matrices A1(t), A2(t), .. ALE), we have: The erponential of anxn matriz A is defined as the follouring: exp A = uhere A° = I. Suppose A(t) and Ä(t) commute. Prove: aexp A(t) = Â(t) exp A(t)

Problem 1 Matrix differentiation - . Let A(t) be an nxn matrir, which depends on t, i.c. its elements are functions of t: (A(0)lij = aij(t), for i, j = 1,...,n The derivative A(t) of A(t) with respect to t, is also an n xn matriz and is defined by taking element-uise derivatives: (Ä(t)ly = åij(t), for i, j = 1,...,n a) For n xn differentiable matrices A1(t), Ag(t) prove: (41(1) A2(t) = Å¡(t)Aa(t) + A1(t)Å½{t) Using induction, prove that for n x n differentiable matrices A1(t), A2(t), .. ALE), we have: The erponential of anxn matriz A is defined as the follouring: exp A = uhere A° = I. Suppose A(t) and Ä(t) commute. Prove: aexp A(t) = Â(t) exp A(t)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question is from subject- Linear System Theory.

Need help on part (b) and part (c) of the problem. part (a) is already solved.

Problem 1 Matrix differentiation -
Let A(t) be an n x n matriz, which depends
on t, i.e. its elements are functions of t:
(A(t)lij = aij(t). for i,j =1,...,n
The derivative A(t) of A(t) with respect to t, is also an n x n matriz and is defined by taking
element-wise derivatives:
LÅ(t)lj = åj(t). for i, j = 1,...,n
For n xn differentiable matrices A1(t), Ag(t) prove:
a)
b)
Using induction, prove that for n x n differentiable matrices A1(t), A2(t), ...
ALE), we have:
...
1
c)
The erponential of a n x n matrix A is defined as the follouring:
еxpA
i!
i-0
where A°
= I. Suppose A(t) and À(t) commute. Prove:
d
exp A(t) = Å(t) exp A(t)

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