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Counterexample

Is this true?

A real matrix with all real eigenvalues is orthogonally diagonalisable

No — it is false.

The counterexample

[[1,1],[0,1]]

Why it fails

Eigenvalue 1 twice but only one eigenvector; orthogonal diagonalisability requires symmetry.

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Execution slip

The idea was right. The computation was not.

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linear algebra

Go deeper· Explained in full in Inner products, orthogonality and the spectral theorem· Asked in CSIR NET June 2025· From Inner Product Spaces and Forms › Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

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