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3. numerical algorithms appear in many components of simulation software for...

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

###### Question details

Numerical algorithms appear in many components of simulation software for quantum physics. The Schr¨odinger equation and others involve complex 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 x ∈ C can be written as x = a + bi, where a,b ∈ R and i = √−1. Suppose we wish to solve A~x = ~ b, but now A ∈ Cn×n and ~x,~ b ∈ Cn. Explain how a linear solver that takes only real-valued systems can be used to solve this equation. Hint: Write A = A1 + A2i, where A1,A2 ∈ Rn×n. Similarly decompose ~x and ~ b. In the end you will solve a 2n×2n real-valued system.