Nilpotent matrix

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In linear algebra, a nilpotent matrix is a square matrix N such that

N^k = 0\,

for some positive integer k. The smallest such k is sometimes called the degree of N.

More generally, a nilpotent transformation is a linear transformation L of a vector space such that Lk = 0 for some positive integer k (and thus, Lj = 0 for all jk). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

Examples[edit]

The matrix

 
M = \begin{bmatrix} 
0 & 1 \\
0 & 0 
\end{bmatrix}

is nilpotent, since M2 = 0. More generally, any triangular matrix with 0s along the main diagonal is nilpotent. For example, the matrix

 
N = \begin{bmatrix} 
0 & 2 & 1 & 6\\
0 & 0 & 1 & 2\\
0 & 0 & 0 & 3\\
0 & 0 & 0 & 0 
\end{bmatrix}

is nilpotent, with


N^2 =   \begin{bmatrix} 
                    0 & 0 & 2 & 7\\
                    0 & 0 & 0 & 3\\
                    0 & 0 & 0 & 0\\
                    0 & 0 & 0 & 0 
                 \end{bmatrix} 

;\ 
N^3 =   \begin{bmatrix} 
                    0 & 0 & 0 & 6\\
                    0 & 0 & 0 & 0\\
                    0 & 0 & 0 & 0\\
                    0 & 0 & 0 & 0 
               \end{bmatrix}

;\ 
N^4 =  \begin{bmatrix} 
                    0 & 0 & 0 & 0\\
                    0 & 0 & 0 & 0\\
                    0 & 0 & 0 & 0\\
                    0 & 0 & 0 & 0 
               \end{bmatrix}.

Though the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example, the matrix

 
N =  \begin{bmatrix} 
                   5 & -3 & 2 \\
                   15 & -9 & 6 \\
                   10 & -6 & 4
               \end{bmatrix}

squares to zero, though the matrix has no zero entries.

Characterization[edit]

For an n × n square matrix N with real (or complex) entries, the following are equivalent:

  1. N is nilpotent.
  2. The minimal polynomial for N is λk for some positive integer kn.
  3. The characteristic polynomial for N is λn.
  4. The only (complex) eigenvalue for N is 0.
  5. tr(Nk) = 0 for all k > 0.

The last theorem holds true for matrices over any field of characteristic 0 or sufficiently large characteristic. (cf. Newton's identities)

This theorem has several consequences, including:

  • The degree of an n × n nilpotent matrix is always less than or equal to n. For example, every 2 × 2 nilpotent matrix squares to zero.
  • The determinant and trace of a nilpotent matrix are always zero.
  • The only nilpotent diagonalizable matrix is the zero matrix.

Classification[edit]

Consider the n × n shift matrix:

S = \begin{bmatrix} 
   0 & 1 & 0 & \ldots & 0  \\
   0 & 0 & 1 & \ldots & 0  \\
   \vdots & \vdots & \vdots & \ddots & \vdots \\
   0 & 0 & 0 & \ldots & 1  \\
   0 & 0 & 0 & \ldots & 0
\end{bmatrix}.

This matrix has 1s along the superdiagonal and 0s everywhere else. As a linear transformation, the shift matrix “shifts” the components of a vector one slot to the left:

S(x_1,x_2,\ldots,x_n) = (x_2,\ldots,x_n,0).

This matrix is nilpotent with degree n, and is the “canonical” nilpotent matrix.

Specifically, if N is any nilpotent matrix, then N is similar to a block diagonal matrix of the form

 \begin{bmatrix} 
   S_1 & 0 & \ldots & 0 \\ 
   0 & S_2 & \ldots & 0 \\
   \vdots & \vdots & \ddots & \vdots \\
   0 & 0 & \ldots & S_r 
\end{bmatrix}

where each of the blocks S1S2, ..., Sr is a shift matrix (possibly of different sizes). This theorem is a special case of the Jordan canonical form for matrices.

For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix

\begin{bmatrix} 
   0 & 1 \\
   0 & 0
\end{bmatrix}.

That is, if N is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b1b2 such that Nb1 = 0 and Nb2 = b1.

This classification theorem holds for matrices over any field. (It is not necessary for the field to be algebraically closed.)

Flag of subspaces[edit]

A nilpotent transformation L on Rn naturally determines a flag of subspaces

 \{0\} \subset \ker L \subset \ker L^2 \subset \ldots \subset \ker L^{q-1} \subset \ker L^q = \mathbb{R}^n

and a signature

 0 = n_0 < n_1 < n_2 < \ldots < n_{q-1} < n_q = n,\qquad n_i = \dim \ker L^i.

The signature characterizes L up to an invertible linear transformation. Furthermore, it satisfies the inequalities

 n_{j+1} - n_j \leq n_j - n_{j-1}, \qquad \mbox{for all } j = 1,\ldots,q-1.

Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.

Additional properties[edit]

(I + N)^{-1} = I - N + N^2 - N^3 + \cdots,
where only finitely many terms of this sum are nonzero.
  • If N is nilpotent, then
\det (I + N) = 1,\!\,
where I denotes the n × n identity matrix. Conversely, if A is a matrix and
\det (I + tA) = 1\!\,
for all values of t, then A is nilpotent.

Generalizations[edit]

A linear operator T is locally nilpotent if for every vector v, there exists a k such that

T^k(v) = 0.\!\,

For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.

References[edit]

  1. ^ R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3

External links[edit]