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Problem 1 Matrix differentiation - on t, i.e. its elements are functions of t: . Let A(t) be an nxn matrir, which depends [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-urise derivatives: [À(t)lij = dij (t). for i, j = 1,...,n a) For n x n differentiable matrices A1(t), Ag(t) prove: (41(t) Aa(t) = Å1(t)A2(t) + A1(t)Ãq{t) b) ALE), we have: Using induction, prove that for nxn differentiable matrices A1(t), A2(t), ... (41(t)A2(t).A(t) = Ả¡(t)A2(t...A ({) + A1 (t)Å2(t)...A (t) + ..- + A (t)A2(t)...«(t) P

Problem 1 Matrix differentiation - on t, i.e. its elements are functions of t: . Let A(t) be an nxn matrir, which depends [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-urise derivatives: [À(t)lij = dij (t). for i, j = 1,...,n a) For n x n differentiable matrices A1(t), Ag(t) prove: (41(t) Aa(t) = Å1(t)A2(t) + A1(t)Ãq{t) b) ALE), we have: Using induction, prove that for nxn differentiable matrices A1(t), A2(t), ... (41(t)A2(t).A(t) = Ả¡(t)A2(t...A ({) + A1 (t)Å2(t)...A (t) + ..- + A (t)A2(t)...«(t) P

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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Problem 1 Matrix differentiation - . Let A(t) be an nxn matrir, which depends on t, i.e. its elements are functions of t: [A(t)lij = aij (t), for i, j = 1,...,n The derivative À(t) of A(t) with respect to t, is also an n x n matrir and is defined by taking element-wrise derivatives: [Ä(t})lij = åij(t), for i, j = 1,...,n a) For nxn differentiable matrices A1(t), Ag(t) prove: (A1(t) A2(t) = Å1()A2(t) + A1(t)Åz(t) Using induction, prove that for nxn differentiable matrices A1(t), A2(t), .., b) AE), we have: (A1(t) A2(t).Ak(t) = Ả¡(t)A2(t)...A (t) + A1(t)Ã2(t)...A (t) + . + A1(t)A2(t)...Å (t) 1 c) The erponential of a n xn matrix A is defined as the follouing: еxpA %3D i! i-0 where A° '= I. Suppose A(t) and A(t) commute. Prove: P a exp A(t) = Á(t) exp A(t)
Transcribed Image Text:Problem 1 Matrix differentiation - . Let A(t) be an nxn matrir, which depends on t, i.e. its elements are functions of t: [A(t)lij = aij (t), for i, j = 1,...,n The derivative À(t) of A(t) with respect to t, is also an n x n matrir and is defined by taking element-wrise derivatives: [Ä(t})lij = åij(t), for i, j = 1,...,n a) For nxn differentiable matrices A1(t), Ag(t) prove: (A1(t) A2(t) = Å1()A2(t) + A1(t)Åz(t) Using induction, prove that for nxn differentiable matrices A1(t), A2(t), .., b) AE), we have: (A1(t) A2(t).Ak(t) = Ả¡(t)A2(t)...A (t) + A1(t)Ã2(t)...A (t) + . + A1(t)A2(t)...Å (t) 1 c) The erponential of a n xn matrix A is defined as the follouing: еxpA %3D i! i-0 where A° '= I. Suppose A(t) and A(t) commute. Prove: P a exp A(t) = Á(t) exp A(t)
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