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In 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 simply an eigenvalue of the graph's adjacency matrix.
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.
In functional analysis, the spectrum of a bounded linear operator A on a Banach spaceFunctional analysis In mathematics, Banach spaces named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions. Definition Banach s is the set of scalars ν such that νI-A does not have a bounded two-sided inverse. Note that by the closed graph theoremIn mathematics, the closed graph theorem is a basic result of functional analysis. For any function T : X → Y we define the graph of T to be the set { x ''y ∈ X ''Y | y T ''x }. The closed graph theorem states the following. Suppose that X and Y, 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.
Occasionally, in an article on matrix theory, one may read a statement like:
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.
Suppose the eigenvalues of a matrix A are λ1,λ2,...,λn. Then the traceIn linear algebra, the trace of an n by n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A i. tr A A + A +. where A represents the i ''j 'th element of A. The use of t of A is λ1+λ2+...+λn and the determinantIn linear algebra, the determinant is a function that associates a scalar det A to every square matrix A''. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants of A is λ1λ2...λn. These two are very important concepts in matrix theory.
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