30x30 Matrix Eigenvalue Norms
Matrix eigenvalues and geometric multiplicities is a medium quant interview question on Linear Algebra, reported to have been seen at Akuna Capital and Goldman Sachs.
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This linear algebra quant interview question is about understanding the eigenstructure of a highly symmetric matrix: one with identical off-diagonal entries and a special value on the diagonal. It forces you to recognize when a matrix can be decomposed into a simple identity part plus a structured low-rank part, and to extract its eigenvalues and multiplicities from that pattern rather than from brute-force computation.
It trains your ability to see rank-one updates, connect them to spectral properties, reason about geometric versus algebraic multiplicities, and manipulate norms of small vectors built from eigenvalues and their multiplicities. This is core quant prep material, sitting at the intersection of spectral theory and matrix intuition.
It matters for quant interviews because many models, covariance matrices, and correlation structures have similar forms. Interviewers want to see that you can exploit matrix structure quickly, avoid mechanical calculations, and use linear algebra to simplify complex-looking problems on the fly.
What it tests
When a matrix is constructed as a sum of a scalar multiple of the identity matrix and a rank-one (or constant) matrix, its eigenvalues can be found by leveraging the structure: the rank-one matrix has a single nonzero eigenvalue (equal to the sum of any row, since all rows are identical), and the rest are zero. Adding a scalar multiple of the identity shifts all eigenvalues by that scalar. This pattern holds because the identity matrix acts as a uniform scaling in every direction, while the rank-one matrix only affects vectors in the direction of its constant row (or column) vector, leaving the orthogonal complement unchanged. The geometric multiplicity of the non-dominant eigenvalue reflects the dimension of the space orthogonal to the constant vector, which is $n-1$ for an $n \times n$ matrix. Recognizing this structure allows you to bypass direct computation and instead use linear algebraic properties to immediately write down the spectrum.
Practise this question with written feedback, or hear it in a spoken mock interview.
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