What Is a Schur Decomposition? (2022)

For large and sparse matrices you can use the (restarted) Arnoldi method to compute a partial Schur decomposition AQ=QR where Q is tall and skinny and R has a few dominant eigenvalues on the diagonal (i.e. eigenvalues on the boundary of the convex hull).<p>MATLAB uses ARPACK&#x27;s implementation of this when you call `eigs`<p>I wrote my own implementation ArnoldiMethod.jl in julia, which unlike MATLAB&#x2F;ARPACK supports arbitrary number types, and also should be more stable in general, and equally fast.<p>[1] <a href="https:&#x2F;&#x2F;github.com&#x2F;JuliaLinearAlgebra&#x2F;ArnoldiMethod.jl">https:&#x2F;&#x2F;github.com&#x2F;JuliaLinearAlgebra&#x2F;ArnoldiMethod.jl</a>

It is a terrible shame that this excellent &quot;What is ..&quot; series is concluded, Higham died earlier this year aged 62.

Remember: if you&#x27;re not certain that your matrix meets the prerequisites for applying the Schur decomposition, you can always apply the Unschur decomposition instead.

explain matrix functions in more detail and how computing a matrix function on an upper triangular matrix is an advantage.<p>how is this a fundamental tenet of numerical linear algebra?<p>also, if I just want the Eigendecomposition, why do I need schur decomposition?<p>The motivation needs more explanation.

This needs a &quot;visual explanation&quot; treatment. The concepts are graspable by the average HN reader, but this presentation is just difficult to approach.<p>I tried using ChatGPT to decompose a lot of the terms, which actually helps, but it&#x27;s hard to vouch for its accuracy: <a href="https:&#x2F;&#x2F;chat.openai.com&#x2F;share&#x2F;709f61b3-b4cb-48df-b97f-fa6877e4b549" rel="nofollow">https:&#x2F;&#x2F;chat.openai.com&#x2F;share&#x2F;709f61b3-b4cb-48df-b97f-fa6877...</a>