Question

Just a general Matlab matrix trick I am trying to understand? What does this line really mean logically?

S=X*X';

What does S accomplish if I transpose any generic matrix against itself?
Thanks

Answer 1

If X is a general NxM matrix, then S=X*X' is the sum of the outer products of each of the columns of X with its transpose. In other words, writing X=[x1,x2,...,xM], S can be written as

S = ∑_i x_i * x_i'

The resulting matrix S is non-negative definite (i.e., eigenvalues are not negative).

If you consider each element in a column of X as a random variable (total N), and the different columns as M independent observations of the N dimensional random vector, then S is the NxN sample covariance matrix (differing by a constant normalization, depending on your conventions) of the rows. Similarly, S=X'*X gives you the MxM covariance matrix of the columns.

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Now if you start restricting the generality and assign special properties to X, then you’ll start seeing patterns emerge for the structure of S. For example, if X is square, has real entries and is orthogonal, then S=I, the identity matrix. If X is square, has complex entries and is a unitary matrix, then S is then again, the identity matrix.

Without knowing the exact circumstances in which this was used in your program, I would assume that they were calculating the covariance matrix.