Search results
Results From The WOW.Com Content Network
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.
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.
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.
A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to and is the only nonzero entry of its column. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain 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.
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]
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.
Minimal polynomial (linear algebra) In linear algebra, the minimal polynomial μA of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μA. The following three statements are equivalent: λ is a root of μA, λ is a root of ...