Wedge Theory / Compound Matrices: Properties and Applications

摘要

The Navy utilizes matrices to analyze radar signals to determine the directionand velocity of aircraft. Matrix analysis is also useful in the sonar classification of submarines. One powerful tool for obtaining information about matrices is wedge theory. (The traditional terminology is compound matrix theory, whereas modern texts speak of mappings on the exterior algebra.) Wedge theory is a fundamental tool in multilinear algebra with important applications to group representations and tensor analysis. Current research indicates that it may also be useful in analyzing noisy data matrices, but this potential has not yet been fully explored. The purpose of this report is to collect details about wedge theory, in one accessible place, to facilitate future exploration of this topic. First, basic properties of the wedge operation are given along with definitions and examples. Then, an application to calculating the rank of a matrix with noise is considered. Finally, since the basic constructions can now be easily implemented on desktop computer algebra systems, the procedures for several such packages are illustrated.