Image Analysis, Classification and Change Detection in Remote Sensing, 4th Edition

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Mathematical Tools

A.1    Cholesky decomposition

Cholesky decomposition is used in some of the routines in this book to solve generalized eigenvalue problems associated with the maximum autocorrelation factor (MAF) and maximum noise fraction (MNF) transformations as well as with canonical correlation analysis. We sketch its justification in the following.

THEOREM A.1

If the p × p matrix A is symmetric positive definite and if the p × q matrix B, where q ≤ p, has rank q, then B^⊤ AB is positive definite and symmetric.

Proof. Choose any q-dimensional vector y ≠ 0 and let x = By. We can write this as

$x=y_1{b}_1+\dots +y_q{b}_q,$

where b_i is the ith column of B. Since B has rank q, we conclude that x ≠ 0 as well, for otherwise the column vectors would be linearly ...