Nilpotent matrix

In linear algebra, a nilpotent matrix is a square matrix N such that M  nilpotent ⟺ there exists  k ∈ N  such that  M k = 0 {displaystyle M{ ext{ nilpotent}}quad iff quad { ext{there exists }}kin mathbb {N} { ext{ such that }}M^{k}=0} In linear algebra, a nilpotent matrix is a square matrix N such that for some positive integer k {displaystyle k} . The smallest such k {displaystyle k} is sometimes called the index of N {displaystyle N} . More generally, a nilpotent transformation is a linear transformation L {displaystyle L} of a vector space such that L k = 0 {displaystyle L^{k}=0} for some positive integer k {displaystyle k} (and thus, L j = 0 {displaystyle L^{j}=0} for all j ≥ k {displaystyle jgeq k} ). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings. For a square matrix M {displaystyle M} , nilpotent is defined as follows.