Nilpotent matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a nilpotent matrix is a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> N such that

N
          
            k
          
        
        =
        0
        
      
    
    {\displaystyle N^{k}=0\,}

for some positive <a href="/facts/Integer/PUwwrawx">integer</a> 
 
 
 
 k
 
 
 {\displaystyle k}
 
. The smallest such 
 
 
 
 k
 
 
 {\displaystyle k}
 
 is called the index of 
 
 
 
 N
 
 
 {\displaystyle N}
 
, sometimes the degree of 
 
 
 
 N
 
 
 {\displaystyle N}
 
.
More generally, a nilpotent transformation is a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> 
 
 
 
 L
 
 
 {\displaystyle L}
 
 of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 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 j\geq k}
 
). Both of these concepts are special cases of a more general concept of <a href="/facts/Nilpotent/rERm2Pp7">nilpotence</a> that applies to elements of <a href="/facts/Ring_(algebra)/JnCUXz63">rings</a>.

Nilpotent matrix open-in-new

Nilpotent matrix