Correspondence of Schur Decomposition and Spectral Decomposition needs P.D?

Question

According to Wikipedia,

[...] the Schur decomposition extends the spectral decomposition. In particular, if $A$ is positive definite, the Schur decomposition of $A$, its spectral decomposition, and its singular value decomposition coincide.

Why is positive definiteness needed in order to make the Schur decomposition correspond to the Spectral decomposition, as opposed to the weaker condition of symmetry?

Thanks

Answer 1

The statement is about three decompositions: the Schur decomposition, the spectral decomposition, and the singular value decomposition. Positive definiteness is needed to make the spectral and singular value decompositions coincide, because the singular values are always nonnegative.

For a Hermitian matrix, the Schur and spectral decompositions coincide, with no hypotheses on the signs of the eigenvalues, and the singular values are the absolute values of the eigenvalues.