B.2. Matrix Computations

Equation solving

Method of least-squares


Singular value decomposition

The singular value decomposition (SVD) gives a similar factorization to the eigendecomposition, but can be applied to non-square matrices. It also gives convenient solutions to find a matrix's rank and null space.

The SVD of an $m \times n$ matrix $A$ takes the form: $$A = U \Sigma V^T$$ where $U$ is an $m \times m$ orthogonal matrix, $V$ is an $n \times m$ matrix with orthogonal columns, and $\Sigma$ is an $m \times m$ matrix with zero entries on all of the off-diagonals, and the diagonal containing only nonnegative entries.

The SVD can be used for many purposes:

  • Calculating matrix rank
  • Calculating a pseudoinverse
  • Dimensionality reduction
  • Least-squares equation solving

In this book we use it primarily for calculating the matrix pseudoinverse $A^+$. Once the SVD is computed, the pseudoinverse $\Sigma^+$ of $\Sigma$ can be computed by taking the reciprocal of all nonzero diagonal entries of $\Sigma$, and leaving the zero entries. Then the pseudoinverse of $A$ is $A^+ = V \Sigma^+ U^T$. It is an exercise to convince yourself that this solution indeed satisfies the properties of the pseudoinverse.

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