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Even though this problem is non-convex in the Euclidean sense and only very locally convex in the Riemannian sense, we discover a structure for this problem that is similar to geodesic strong convexity, namely, weak-strong convexity. This allows us to apply similar arguments from convex optimization when studying the convergence of the steepest descent algorithm but with initialization conditions that do not depend on the eigengap <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\delta $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b4<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. When <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\delta &gt;0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b4<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we prove exponential convergence rates, while otherwise the convergence is algebraic. Additionally, we prove that this problem is geodesically convex in a neighbourhood of the global minimizer of radius <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {O}}(\\sqrt{\\delta })$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msqrt>\n                      <mml:mi>\u03b4<\/mml:mi>\n                    <\/mml:msqrt>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10957-024-02538-8","type":"journal-article","created":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T19:01:39Z","timestamp":1728414099000},"page":"920-959","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Steepest Descent"],"prefix":"10.1007","volume":"203","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2376-5578","authenticated-orcid":false,"given":"Foivos","family":"Alimisis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bart","family":"Vandereycken","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,10,8]]},"reference":[{"issue":"2","key":"2538_CR1","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1023\/b:acap.0000013855.14971.91","volume":"80","author":"P-A Absil","year":"2004","unstructured":"Absil, P.-A., Mahony, R., Sepulchre, R.: Riemannian geometry of Grassmann manifolds with a view on algorithmic computation. 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