Linear Algebra › Section G · Symmetric matrices and quadratic forms › Topic G.4
The singular value decomposition
Not every matrix can be diagonalised, and a rectangular matrix has no eigenvalues at all. The singular value decomposition factors every matrix into a rotation, a stretch along perpendicular axes and another rotation, and it is one of the most useful factorisations in applied mathematics.
In this topic
- From eigenvalues to singular values
- Singular values
- Singular values reveal the rank
- The decomposition
- Worked example: a square matrix
- Worked example: a tall matrix
- Bases for the four fundamental subspaces
- Condition number and the pseudoinverse
- The geometric picture
- Common mistakes to avoid
- Key takeaways
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