Shifting Eigenvalues

A linear algebra question asked in interview test at Millenium

Tip of the week: Linear Algebra is one of the fundamental questions asked in quant interview for global HFT firms and hedge funds, as it underpins many algorithms used in financial modeling and trading strategies. Therefore, preparing extensively in linear algebra is vital for interviews at these firms, where demonstrating proficiency can significantly impact one's ability to contribute to and innovate in this high-stakes, technically demanding field.

Let A be a n X n matrix with eigenvalues 5 and 7. Find the sum of the eigenvalues of

\( B = (({A - 3I_n)^{-1})}^3\)

Try to solve this problem yourself before moving on to the solution below

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Solution

The eigenvalues of A - 3I_n are 2 and 4, as the -3I_n decreases both eigenvalues by 3. Then, (A - 3I_n)^{-1} has eigenvalues 0.5 and 0.25, as the inverse has eigenvalues that are the reciprocal of the original eigenvalues.

Lastly, ((A - 3I_n)^{-1})^3 would have eigenvalues ( 0.5 )^3 = 0.125 and (0.25)^3 = 1/64 , as we would cube the eigenvalues. Adding these up, we get that our answer is 9/64

\(\frac{1}{8} + \frac{1}{64} = \frac{9}{64}\)

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😃 Over to you: Were you able to solve this?

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