Is it possible to compute e for some matrix A E Rx2? Mathematically, does such an...
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Is it possible to compute e for some matrix A E R²x2? Mathematically, does such an operation make any sense? Consider that complex number a + bi can be interpreted geometrically as a scaling and rotation operation in the imaginary-real plane, where radius r is a scaling factor and is the rotation angle. Recall the rotation matrix R that rotates a vector in the imaginary-real plane counterclockwise through an angle with respect to the positive real (Re) axis about the origin. cos o sin o Scaling by r, we have R = - sin cos r cos o r sin p Letting a = r cos [9] a rR= rR= -r sin r cos o and b = r sin o we have Im 0 cose e¹= cos o + i sin o sin o Re From Euler's formula, eiπ = cos π + i sin π, we can express ei as eª with A=[5] Α 0 Now recall the Maclaurin series for the exponential function eº is Σx=0 ml Maclaurin series to approximate e¹"? What will we find? = 1+x+ = N8 + What if we substitute A into the Write a Python program to approximate the matrix representation of e*. Use NumPy methods to compute powers of A in the Maclaurin series. In your program, use a while iterative construct to continue accumulating terms in the series until the approximation does not change between the approximation with N terms and N + 1 terms. If you denote the approximation with N terms as SN and the approximation with N + 1 terms as SN+1, you can stop iterating if the matrix norm ||SN+¹ – SN ||2 < T for some small 7, for example, 10-¹7. Is it possible to compute e for some matrix A E R²x2? Mathematically, does such an operation make any sense? Consider that complex number a + bi can be interpreted geometrically as a scaling and rotation operation in the imaginary-real plane, where radius r is a scaling factor and is the rotation angle. Recall the rotation matrix R that rotates a vector in the imaginary-real plane counterclockwise through an angle with respect to the positive real (Re) axis about the origin. cos o sin o Scaling by r, we have R = - sin cos r cos o r sin p Letting a = r cos [9] a rR= rR= -r sin r cos o and b = r sin o we have Im 0 cose e¹= cos o + i sin o sin o Re From Euler's formula, eiπ = cos π + i sin π, we can express ei as eª with A=[5] Α 0 Now recall the Maclaurin series for the exponential function eº is Σx=0 ml Maclaurin series to approximate e¹"? What will we find? = 1+x+ = N8 + What if we substitute A into the Write a Python program to approximate the matrix representation of e*. Use NumPy methods to compute powers of A in the Maclaurin series. In your program, use a while iterative construct to continue accumulating terms in the series until the approximation does not change between the approximation with N terms and N + 1 terms. If you denote the approximation with N terms as SN and the approximation with N + 1 terms as SN+1, you can stop iterating if the matrix norm ||SN+¹ – SN ||2 < T for some small 7, for example, 10-¹7.
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Cost Management Accounting and Control
ISBN: 978-0324559675
6th Edition
Authors: Don R. Hansen, Maryanne M. Mowen, Liming Guan
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