Matrix Eigenvalues Squared Sum Hunt

This linear algebra interview question focuses on eigenvalues of a two-by-two matrix and, more specifically, on a symmetric function of those eigenvalues. It asks you to connect the spectrum of the matrix to simple quantities that can be read directly from its entries. In a quant prep context, it is a neat example of turning an apparently spectral question into a problem about basic matrix invariants.

It trains your understanding of how eigenvalues relate to trace and determinant, and how to manipulate expressions built from eigenvalues without explicitly computing them. You practice mental algebra, comfort with characteristic polynomials, and the ability to recognize shortcuts that exploit structure. This is core linear algebra fluency for quant interviews.

This matters for quant interviews because many modeling, pricing, and risk problems boil down to matrix analysis under time pressure. Interviewers want to see you quickly extract key properties of a matrix, avoid unnecessary computation, and use clean theoretical relationships. Problems like this separate candidates who merely remember formulas from those who can apply linear algebra intuitively in real quant interview settings and more advanced quant prep.

What it tests

For any square matrix, the eigenvalues are the roots of its characteristic polynomial, which is constructed by subtracting $\lambda$ times the identity matrix from the original matrix and taking the determinant. The coefficients of this polynomial are deeply connected to the matrix's trace and determinant: specifically, for a $2 \times 2$ matrix, the sum of the eigenvalues equals the trace, and the product equals the determinant. The sum of the squares of the eigenvalues can be found without explicitly solving for them, since it equals $(\text{trace})^2 - 2 \times (\text{determinant})$. This relationship arises from expanding $(\lambda_1 + \lambda_2)^2 = \lambda_1^2 + 2\lambda_1\lambda_2 + \lambda_2^2$ and rearranging in terms of trace and determinant. Recognizing these relationships allows you to compute symmetric functions of the eigenvalues directly from the matrix entries, bypassing the need to solve the characteristic equation in full.