Sums of Projections

In school I learned a notion of projection that allowed one to prove things about linear transformations, aka matrices. I recount here some of the theorems but probably with too few hypotheses. We hold the field and space X fixed and consider linear transformations (LTs) of X to X. A projection E is a LT such that E = EE. 0 and I are projections and any diagonal matrix all of whose diagonal elements are either 0 or 1 is a projection and there are more. The same transformation in another coordinate system is also a projection and is seldom diagonal. Any hermitian matrix can be written as linear combination of projections, one per eigenvalue.

For any unit vector x there is a projection E_x such that E_xx = x and E_xy = 0 if x and y are orthogonal. This suggests that projections are useful in inner product spaces. If λ_i are the eigenvalues of M and E_i are the respective projections for the corresponding eigenvector them M = Σ λ_iE_i. If x_i are the components of an eigenvector then its projection has elements x_ix_j. Much of this, I fear, is limited to hermitian matrices.

One use of projections is as follows: Use one of the methods that finds an eigenvector of a matrix M with a large eigenvalue. Subtract the corresponding projection from M. The new M has the same eigenvector but its eigenvalue is now 0 and you can iterate finding the eigenvector with the largest eigenvalue of the new matrix.

I hope soon to have some eigenvectors of general matrices and test imperially how much of this logic works for non hermitian matrices.

In an inner product space (denoted by juxtaposition of vector expressions) for every vector v≠0 there is a projection E so that for every vector x, Ex = ((vx)/(vv))v.

If {b_i} forms a basis for our space and b_ib_j = δ_ij we call this an orthogonal coördinate system. If {v_i} are scalars and v = Σv_ib_i then the matrix E_ij = (v_iv_j)/(vv) is the E that goes with the v above.

Many years ago I wrote a program that repeatedly squared a matrix thus squaring its eigenvalues until the eigenvalue that had had the largest magnitude dominated which meant that the matrix was essentially a projection times some large scalar. Any row of the matrix could then be taken as an eigenvector and its original eigenvalue easily computed. The corresponding projection could be deducted from the original matrix and the next largest eigenvalue, and its eigenvector could be found etc.

To avoid exponent overflow, or underflow, I did a scalar normalization towards unity. This failed for many real matrices because they had complex roots and the high-power matrix was not nearly a projection. The projection I sought was a matrix with complex coefficients and my program dealt only with real matrices; I needed symmetry breaking to get off of the real line.