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the norms of a vector X = [1,12, ...,I" and matrix A = (a5) are respectively defined as ||X|| = E and ||A| - max laisl- j=1,2, i=1 Alternatively we can define

the norms of a vector X = [1,12, ...,I" and matrix A = (a5) are respectively defined as ||X|| = E and ||A| - max laisl- j=1,2, i=1 Alternatively we can define

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 24EQ
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the norms of a vector X = [1, 12, ...,In" and matrix A = (a;5) are respectively defined as ||X || = Ei and ||A|| = max laisl. %3D j=1,2,.n i=1 Alternatively we can define ||X || = and ||A|| max j=1,2, i=0 i=1 The latter definition means that the norm of a matrix A is defined as the maximum of the sums obtained by addition of the absolute values of the entries in each column of A. Now consider the system i1 = trı + 2tr2 + t iz = (sin t)r1 +-r2 + 2r3 +e i3 = 2(cos t)r1 + 2(sin t)r2 ++e. Assume there exist solutions X and Y on the interval (1,4) with X (2) = [3, 1.9, 4" and Y (2) = [3.2, 2.1, 3.9|". Use alternative definitions given above and Gronwall's inequality to estimate the error ||X (t) – Y (t)|| for 1<t<4.
Transcribed Image Text:the norms of a vector X = [1, 12, ...,In" and matrix A = (a;5) are respectively defined as ||X || = Ei and ||A|| = max laisl. %3D j=1,2,.n i=1 Alternatively we can define ||X || = and ||A|| max j=1,2, i=0 i=1 The latter definition means that the norm of a matrix A is defined as the maximum of the sums obtained by addition of the absolute values of the entries in each column of A. Now consider the system i1 = trı + 2tr2 + t iz = (sin t)r1 +-r2 + 2r3 +e i3 = 2(cos t)r1 + 2(sin t)r2 ++e. Assume there exist solutions X and Y on the interval (1,4) with X (2) = [3, 1.9, 4" and Y (2) = [3.2, 2.1, 3.9|". Use alternative definitions given above and Gronwall's inequality to estimate the error ||X (t) – Y (t)|| for 1<t<4.
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