Some Basic Matrix Theorems - Quandt.com

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some Basic Matrix TheoremsRichard E. QuandtPrinceton UniversityDefinition a square Matrix of ordernand let be a scalar quantity. Then det(A I)is called the characteristic polynomial is clear that the characteristic polynomial is annthdegree polynomial in and det(A I) = 0will haven(not necessarily distinct) solutions for .Definition values of that satisfy det(A I) = 0 are the characteristic roots oreigenvalues follows immediately that for each that is a solution of det(A I) = 0 there exists a nontrivialx( ,x6= 0) such that(A I)x=0.(1)Definition vectorsxthat satisfy Eq.(1) are the characteristic vectors or eigenvectors consider a particular eigenvalue and its corresponding eigenvectorx, for which we have x=Ax.

Regression Theory 3 Theorem 3. If λi is a repeated root with multiplicity m >= 2, then there exist m orthonormal eigenvectors corresponding to λi. Proof. First, we note that corresponding to λi there will be at least one eigenvector xi.For any arbitrary nonzero vector xi one can always find an additional n−1 vectors yj, j =2,...,n, so that xi, together with the n−1 y-vectors …

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