First, yes, I overgeneralized. Matrices don't represent natural language and logic well.

But, the kinds of problems you're talking about--music analysis, picture analysis, and anything you eventually want to put into PCA--are perfect for matrix methods. It's popular to start music and picture analysis with a discrete Fourier transform, which is a matrix operation. Or you use MPEG, which is all matrices. Or you construct feature detectors, say edge detectors or contrast detectors, using simple neural networks such as those found in primary visual cortex, and you implement them with matrices. Then you pass those into higher-order feature detectors, which also use matrices. You may break information out of the matrices and process it logically further downstream, but that will be downstream of PCA. As a general rule, PCA is used only on data that has so far existed only in matrices. Things that need to be broken out are not homogenous enough, or too structured, to use PCA on.

There's an excellent book called Neural Engineering by Chris Eliasmith in which he develops a matrix-based programming language that is supposed to perform calculations the way that the brain does. It has many examples of how to tackle "intelligent" problems with only matrices.