Schur decomposition applications

Question

I know that Schur decomposition is important in matrix theory and linear algebra.

I am doing a research and wondering:

1. Why is it that important?

2. What are some applications of it outside the math world like in engineering, computer science etc. ?

Answer 1

The Schur decomposition $A = USU^T$ of a dense matrix $A$ can be obtained using a sequence of orthogonal similarity transformations. This allows us to obtain the eigenvalues of the matrix $A$ in a numerically reliable manner. Moreover, the eigenvectors of $A$ can be computed by substitution and backtransformation from the (quasi) triangular matrix $S$. It is possible to prevent overflow in the substitution, making the entire process remarkably robust.

A large number of physical systems can be modeled using linear time-invariant dynamical systems \begin{align} \dot{x} &= A x + B u(t) \\ y &= Cx + Du(t), \end{align} where $x$ describes the state of the system, $u$ is the control and we are primarily interested in the output $y$. A large scale example is a weather model, where the state vector $x$ includes the pressure, temperature, humidity and wind velocity at every grid point. Typically, the dimension is so large that direct integration is too time consuming. The successful application of model reduction by balanced truncation hinges on the matrix $A$ being (asymptotically) stable. Computing the all eigenvalues is the only way to be sure.

Buildings are often modeled using a system of ordinary differential equation of the type $$M \ddot{x} + D \dot{x} + Kx = F(t)$$ Here $x$ is a vector of displacements of nodes in the building and $F$ is an external force, say, an Earth quake. It is not uncommon that the damping matrix $D$ is a linear combination of the mass matrix $M$ and the stiffness matrix $K$. In order to understand the response of the building to the external force, one needs to solve the generalize eigenvalue problem $$Mv = \lambda K x.$$ There is a generalized Schur form which is relevant here.

Dense eigensolvers are used as kernels for sparse codes such as subspace iteration with Ritz acceleration or for the construction of preconditioners for sparse linear systems.

Answer 2

Why look outside the math world? Here is simple-enough multi-variable calculus problem that required the Schur decomposition.

Here is an application from Statistics, one that is applied in the derivation of the Kalman filter.

See the comments on that at the end of Jean Marie’s answer here.

A complementary view ( is to interpret matrix A as a covariance matrix of a certain multivariate normal random variable in $\mathbb{R^3}$ and B as the covariance matrix associated to a marginal distribution in $\mathbb{R^2}$

And needless to say, some others from dynamical systems and control ( already covered by Carl Christian’s answer)