A Detour into Schur and Eigen Decomposition
Team Latte
Aug 19, 2005

We have talked about Cholesky decomposition in some detail and tried to understand - in as conceptual way as possible - what is a cholesky matrix in the context of portfolio risk. There are two other kinds of decomposition applicable in the case of square variance-covariance (VCV) matrices that have very limited application in portfolio risk analysis.

These decomposition methods are not widely used in practice, and though they can be easily applied mathematically to any VCV matrix, we find that getting an intuitive feel for these processes is difficult. Both these methods, described very briefly below, can be used in place of cholesky decomposition. Eigen decomposition, in fact is used by quite a few quantitative risk managers to apply Monte Carlo simulation technique to do value at risk (VaR) analysis as well as fixed income "quants" to analyze correlations between various interest rate (on the yield curve) movements.

Schur Decomposition

So, what is this Cholesky matrix?

Schur decomposition is a method to decompose a square n x n matrix into an upper triangular matrix. This upper triangular matrix can be expressed as the sum of a diagonal matrix and a strictly upper triangular matrix. Schur decomposition can be used to calculate the eigenvalues of a defective or non-diagonalizable matrix - e.g. a matrix with double eigenvalues.

Eigen Decomposition

Eigen decomposition is a method to decompose a matrix A (n x n square matrix) into a diagonal matrix, D. This is achieved by pre-multiplying the matrix A by the inverse of a matrix P and then post-multiplying it by the matrix P. The diagonal elements of the resulting diagonal matrix D are the eigen values of matrix A. And the matrix P is the matrix of the eigen vectors of A.

As a result of this decomposition, we can express the matrix A as the product of the matrix multiplication of P, D and P inverse.

The benefit of this decomposition arises because of the properties of diagonal matrix D. Using this transformation we can easily calculate the power of a matrix; we only need to calculate the power of the diagonal matrix. This calculation is trivial; essentially the power of the diagonal elements of the diagonal matrix. And then pre-multiply the result by the matrix P and then post-multiply it by the inverse of matrix P.

However it should be kept in mind that in order for the diagonalization to be possible the matrix A and P must be square matrices.

Disclaimer
"Risk Latte uses proprietary and non-proprietary mathematical and empirical models to measure the volatility and estimate the direction of the market. There is no guarantee of any particular outcome happening and readers must exercise caution while interpreting the conclusions of this article. Risk Latte Company is not a registered stock broker or an SFC registered entity and readers must take advise from their financial advisors, stock brokers, research analysts and bankers while making any buying or selling decisions. Risk Latte Company is not in the business of making stock or asset forecasts whether explicitly or implicitly and shall not be responsible for and/or liable for any losses arising out of any trading decisions based on the above article."

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