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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.)
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.
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.
For example, in ordinary least squares, the regression problem is to choose a vector β of coefficient estimates so as to minimize the sum of squared residuals (mispredictions) e i: in matrix form, Minimize ( y − X β ) T ( y − X β ) {\displaystyle (y-X\beta )^{\textsf {T}}(y-X\beta )}
Write the triangular matrix U as U = D + N, where D is diagonal and N is strictly upper triangular (and thus a nilpotent matrix). The diagonal matrix D contains the eigenvalues of A in arbitrary order (hence its Frobenius norm, squared, is the sum of the squared moduli of the eigenvalues of A , while the Frobenius norm of A , squared, is the ...
Since L and M commute, the matrix L + M is nilpotent and I + (L + M)/2 is invertible with inverse given by a Neumann series. Hence L = M. If A is a matrix with positive eigenvalues and minimal polynomial p(t), then the Jordan decomposition into generalized eigenspaces of A can be deduced from the partial fraction expansion of p(t) −1.
Nilpotent matrices with complex entries form the main motivating case for the general theory, corresponding to the complex general linear group.From the Jordan normal form of matrices we know that each nilpotent matrix is conjugate to a unique matrix with Jordan blocks of sizes …, where is a partition of n.