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Question: numerical algorithms appear in many components of simulation software for...

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Numerical algorithms appear in many components of simulation software for quantunm physics. The Schrödinger equation and others involve complez numbers in C, however so we must extend the machinery we have developed for solving linear systems of equations to this case. Recall that a complex number z E C can be written as x = a + bi, where a, b є R and-VT. Suppose we wish to solve Ar = b, but now A Cn>< n and 2, b C. Explain how a linear solver that takes only real-valued systems can be used to solve this equation Hint: Write A-A1 + Agi, where Ai, A2 є Rnxn. Similarly decompose z and b. In the end you will solve a 2n x 2n real-valued system. Suppose A Rnxn is invertible. Show that A-1 can be obtained via Gaussian elimi- nation on augmented matrix (시1nxn ).

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