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nilpotent matrix is always less than or equal to. n {\displaystyle n} For example, every. 2 × 2 {\displaystyle 2\times 2} nilpotent matrix squares to zero. The determinant and trace of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible. The only nilpotent diagonalizable matrix is the zero matrix.
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
Nilpotent. In mathematics, an element of a ring is called nilpotent if there exists some positive integer , called the index (or sometimes the degree), such that . The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras. [1]
Eigenvalues and eigenvectors. In linear algebra, an eigenvector (/ ˈaɪɡən -/ EYE-gən-) or characteristic vector is a vector that has its direction unchanged by a given linear transformation. More precisely, an eigenvector, , of a linear transformation, , is scaled by a constant factor, , when the linear transformation is applied to it: .
Min-max theorem. In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature.
Variational method (quantum mechanics) In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. [1] The basis for this method is the variational principle. [2][3]
In mathematics, a unipotent element[ 1 ]r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1) n is zero for some n. In particular, a square matrix M is a unipotent matrix if and only if its characteristic polynomial P (t) is a power of t − 1. Thus all the eigenvalues of a unipotent matrix are 1.
The operator T is not nilpotent: take f to be the function that is 1 everywhere and direct calculation shows that T n f ≠ 0 (in the sense of L 2) for all n. However, T is quasinilpotent. First notice that K is in L 2 (X, m), therefore T is compact. By the spectral properties of compact operators, any nonzero λ in σ(T) is an eigenvalue.