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)Ž{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)Ž{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.
Transcribed Image Text: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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