In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} ...

See nilpotent matrix for more. In the factor ring Z / 9 Z {\displaystyle \mathbb {Z} /9\mathbb {Z} } , the equivalence class of 3 is nilpotent because...

states that any matrix is uniquely the sum of a diagonalizable matrix and a nilpotent matrix. Over an algebraically closed field, diagonalizable matrices...

triangular matrix are also 0, the matrix is called strictly (lower or upper) triangular. All finite strictly triangular matrices are nilpotent of index...

The matrix PX is idempotent. More generally, the trace of any idempotent matrix, i.e. one with A2 = A, equals its own rank. The trace of a nilpotent matrix...

the zero matrix. Identity matrix, the multiplicative identity for matrices Matrix of ones, a matrix where all elements are one Nilpotent matrix Single-entry...

i.e. when T is a square matrix (Nilpotent matrix) with complex entries, σ(T) = {0} if and only if T is similar to a matrix whose only nonzero entries...

The adjacency matrix can be used to determine whether or not the graph is connected. If a directed graph has a nilpotent adjacency matrix (i.e., if there...

{e^{4}+1}{2e}}\\\end{bmatrix}}.} A matrix N is nilpotent if Nq = 0 for some integer q. In this case, the matrix exponential eN can be computed directly...

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...

inverse B − 1 {\displaystyle B^{-1}} and N {\displaystyle N} is a nilpotent matrix, then A D = [ B − 1 0 0 0 ] {\displaystyle A^{D}={\begin{bmatrix}B...

(the identity matrix), for any integer a between 0 and n inclusive. If N is any nilpotent matrix, then N is similar to a block diagonal matrix of the form...

In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has...

with the group of unit quaternions. The Heisenberg group is a connected nilpotent Lie group of dimension ⁠ 3 {\displaystyle 3} ⁠, playing a key role in...

with M nilpotent, commuting with N and hence L. But then 0 = S2 − T2 = 2(L − M)(I + (L + M)/2). Since L and M commute, the matrix L + M is nilpotent and...

Jordan block corresponding to λ is of the form λI + N, where N is a nilpotent matrix defined as Nij = δi,j−1 (where δ is the Kronecker delta). The nilpotency...

the Jordan normal form of matrices we know that each nilpotent matrix is conjugate to a unique matrix with Jordan blocks of sizes λ 1 ≥ λ 2 ≥ … ≥ λ r , {\displaystyle...

matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...

similarly. A nilpotent element is an element a such that an = 0 for some n > 0. One example of a nilpotent element is a nilpotent matrix. A nilpotent element...

then so is D and D−1(PAP−1) is equal to the identity plus a nilpotent matrix. But such a matrix is always invertible (if Nk = 0 the inverse of 1 − N is 1...

1&0\\0&0\end{bmatrix}}.} For n ≥ 2, the matrix ring Mn(R) over a nonzero ring has zero divisors and nilpotent elements; the same holds for the ring of...

zero matrix, corresponding to the dual number ε {\displaystyle \varepsilon } . Generally, if ε {\displaystyle \varepsilon } is a nilpotent matrix, then...

Jordan–Chevalley decomposition which expresses a matrix as a sum of a diagonalizable and a nilpotent part. A Hermitian matrix has all real eigenvalues and can always...

splitting of A. Gauss–Seidel method Jacobi method List of matrices Nilpotent matrix Successive over-relaxation Burden & Faires (1993, p. 404) Isaacson...

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 mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities...

geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module K [ T ] / T 2 , {\displaystyle K[T]/T^{2},} showing algebraic...

More generally, all nilpotent groups are solvable. In particular, finite p-groups are solvable, as all finite p-groups are nilpotent. In particular, the...

whose elements are nilpotent elements. 3.  A nilpotent ideal is an ideal whose power Ik is {0} for some positive integer k. Every nilpotent ideal is nil, but...

commuting nilpotent matrix and diagonalizable matrix Schur decomposition — similarity transform bringing the matrix to a triangular matrix Singular value...