Some Basic Matrix Theorems - Quandt.com

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.(2)Premultiply (2) by an arbitrary nonsingular matrixPwe obtain P x=PAx=PAP 1Px,(3)and definingPx=y, y=PAP 1y.(4)Hence is an eigenvalue andyis an eigenvector of the matrixPAP matricesAandPAP 1are called similar have shown above that any eigenvalue ofAis also an eigenvalue ofPAP show the converse, , that any eigenvalue ofPAP 1is also an eigenvalue matrixAis symmetric ifA=A.

2 Quandt Theorem 1. The eigenvalues of symmetric matrices are real. Proof. A polynomial of nth degree may, in general, have complex roots. Assume then, contrary to the assertion of the theorem, that λ is a complex number. The corresponding eigenvector x may have one or more complex elements, and for this λ and this x we have Ax = λx. (5)

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