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For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix []. That is, if is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b 1, b 2 such that Nb 1 = 0 and Nb 2 = b 1. This classification theorem holds for matrices over any field. (It is not necessary for the field to be algebraically closed.)
The Jordan block corresponding to λ is of the form λI + N, where N is a nilpotent matrix defined as N ij = δ i,j−1 (where δ is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where f is a complex analytic function. For example, in principle the Jordan form could give a closed-form expression for the ...
Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with n = 2 {\displaystyle n=2} ). Both are linked, also through supersymmetry and Morse theory , [ 6 ] as shown by Edward Witten in a celebrated article.
If a, b, c, are real numbers (in the ring R), then one has the continuous Heisenberg group H 3 (R).. It is a nilpotent real Lie group of dimension 3.. In addition to the representation as real 3×3 matrices, the continuous Heisenberg group also has several different representations in terms of function spaces.
There is also a real Schur decomposition. If A is an n × n square matrix with real entries, then A can be expressed as [4] = where Q is an orthogonal matrix and H is either upper or lower quasi-triangular. A quasi-triangular matrix is a matrix that when expressed as a block matrix of 2 × 2 and 1 × 1 blocks is
The matrix exponential of another matrix (matrix-matrix exponential), [24] is defined as = = for any normal and non-singular n×n matrix X, and any complex n×n matrix Y. For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y , because the multiplication operator for matrix ...
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix , which is involved in the closed-form solution of systems of linear differential equations .
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.