Implement matrix transforms and discuss eigenvalues

Q: Implement matrix transforms and discuss eigenvalues

This question evaluates understanding of matrix manipulation and numerical linear algebra, including in-place matrix transforms and rotations, sparse matrix representations, eigenvalue computation, and their extensions to higher-order tensors.

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Question

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You are given a square matrix A (size n × n).

Diagonal symmetry (transpose): Transform A into its transpose (mirror across the main diagonal). Prefer an in-place algorithm when possible.
Rotate 90 degrees: Rotate the matrix by 90° clockwise (again, ideally in-place).
Eigenvalues: Explain how you would compute the eigenvalues of A .

Follow-ups

If A is a sparse matrix (most entries are zero), how would your representation and approach change for the above operations (especially eigenvalue computation)?
If the data is 3D (a tensor / 3D array), how do “transpose/diagonal symmetry” and “rotation” generalize, and what does an eigenvalue-like concept become?

Assume typical interview constraints such as n up to a few thousand for dense operations (memory permitting), and much larger for sparse matrices.

Implement matrix transforms and discuss eigenvalues

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Follow-ups

Solution

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