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ArnoAtWork

Eigenvectors

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Hello, Sometimes, when I calculate the eigenvectors of vertex cloud, I find 1 or 2 eigenvalues. So, in my OBB determination, to have 1 value means world axes(X,Y,Z). But when I only find 2 values....what can I do? Thanks a lot.

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When you have an eigenvalue of algebraic and geometric multiplicity of 2, the corresponding eigenspace is two dimensional. This space happens to be the orthogonal complement of the one-dimensional eigenspace for the other distinct eigenvalue. Any pair of unit-length and orthogonal vectors in the two-dimensional eigenspace is a valid choice.

For example, suppose your covariance matrix is Diagonal(1,1,2). An eigenvector for eigenvalue 2 is (0,0,1). Eigenvectors for eigenvalue 1 are all of the form (x,y,0), where x^2+y^2=1. The infinitely many pairs of orthogonal eigenvectors are (cos(t),sin(t),0) and (-sin(t),cos(t),0) for any angle t in [0,2*pi).

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