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EigenvalueIn linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. The vector x is called an eigenvector. In matrix theory, an element in the underlying ring R of a square matrix A is called a right eigenvalue if there exists a nonzero column vector x such that Ax=λx, or a left eigenvalue if there exists a nonzero row vector y such that yA=yλ. If R is commutative, the left eigenvalues of A are exactly the right eigenvalues of A and are just called eigenvalues. If R is not commutative, e.g. quaternions, they may be different. In graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix A, or (increasingly) of the graph's Laplacian matrix , where T is a diagonal matrix holding the degree of each vertex, and in , 0 is substituted for . The word eigenvalue comes from the German Eigenwert which means "proper or characteristic value." Multiplicity Suppose A is a square matrix over a commutative ring. The algebraic multiplicity (or simply multiplicity) of an eigenvalue λ of A is the number of factors t-λ of the characteristic polynomial of A. The geometric multiplicity of λ is the number of factors t-λ of the minimal polynomial of A or equivalently the nullity of (λI-A). An eigenvalue of algebraic multiplicity 1 is called a simple eigenvalue. Spectrum In functional analysis, the spectrum of a bounded linear operator A on a Banach space is the set of scalars ν such that νI-A does not have a bounded two-sided inverse. Note that by the closed graph theorem, if a bounded operator has an inverse, the inverse is necessarily bounded. If the underlying Banach space is finite dimensional, then the spectrum of A is the same of the set of eigenvalues of A. This follows from the fact that on finite dimensional spaces injectivity of a linear operator A is equivalent to surjectivity of A. Multiset of eigenvalues Occasionally, in an article on matrix theory, one may read a statement like: - The eigenvalues of a matrix A are 4,4,3,3,3,2,2,1.
It means the algebraic multiplicity of 4 is two, of 3 is three, of 2 is two and of 1 is one. This style is used because algebraic multiplicity is the key to many mathematical proofs in matrix theory. Trace and determinant Suppose the eigenvalues of a matrix A are λ1,λ2,...,λn. Then the trace of A is λ1+λ2+...+λn and the determinant of A is λ1λ2...λn. These two are very important concepts in matrix theory. See also Please refer to eigenvector for some other properties of eigenvalues.
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