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Evaluate the convergence of the algorithms:
From the convergence proof of power method, LR and QR algorithm for the computation of eigenvalues we see that the easiest case to proof convergence of these algorithms is when all eigenvalues of a matrix are distinct and their absolute values are also distinct.Conversely, it is not difficult to imagine that the convergence can be difficult to obtain when several eigenvalues have similar absolute values or in the case of repeated eigenvalue. In this project, we attempt to examine some of these more challenging cases.Algorithmic Analysis(a) Show that for any real valued matrix A, if a complex number is an eigenvalue, the complex conjugate μ must also be an eigenvalue. (b) Consider a matrix A with a complex eigenvalue with non-zero imaginary part. Consider the Jornal canonical form of matrix A obtained via similarity transformation. What are the relationships between elementary Jordan blocks associated with and ?(c) When using the power method or the LR or QR algorithm, can the algorithm converge to an upper-triangular matrix?(d) Propose a possible approach to compute complex eigenvalues of a real valued matrix A.Computer Implementation(a) Implement LR and QR for computation of eigenvalues including algorithm to first transform the input matrix to a Henssenberg matrix.(b) Validate the correctness of your implementation.(c) Evaluate the convergence of the algorithms in the case of matrix with complex eigenvalue.
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Juan is g years old and Eva is 2 years younger than Juan. a.Find the sum of their ages in terms of g. b.Find the sum of their ages in g years'' time,in terms of g.
Differance between Expanded Notation vs. Standard Notation ? A number written in expanded notation is broken down into parts just like it is in a place-value table. Example
Which general famously stated 'I shall return'? A. Bull Halsey B. George Patton C. Douglas MacArthur D. Omar Bradley
3x+y=9 5x-y=7
Change of base: The final topic that we have to look at in this section is the change of base formula for logarithms. The change of base formula is,
One-to-one Correspondence : Suppose you are given a certain number of cups and saucers, and are asked to find out whether there are enough saucers for all the cups. How would you
what is the perimeter of a rhombus
Solving an equation using Multiplication and Division A variable is a symbol that represents a number. Usually we use the letters like n , t , or x for variables. For
calculate the area of a trapezoid with height 8cm base 18cm and 9cm
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