What is a nilpotent linear transformation?
What is a nilpotent linear transformation?
More generally, a nilpotent transformation is a linear transformation of a vector space such that for some positive integer (and thus, for all. ). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.
How do you prove a linear transformation is nilpotent?
The element T ∈ A(V) is called nilpotent if Tm = 0 for some positive integer m. If T ∈ A(V) is nilpotent, then k is called the index of nilpotence of T if Tk = 0 but Tk-1 = 0. A square matrix A is termed nilpotent if Am = 0 for some positive integer m, and of index k if Ak = 0 but Ak-1 = 0.
What is a nilpotent linear operator?
In operator theory, a bounded operator T on a Hilbert space is said to be nilpotent if Tn = 0 for some n. It is said to be quasinilpotent or topologically nilpotent if its spectrum σ(T) = {0}.
Is nilpotent zero a matrix?
So no power of A can be the zero matrix. (b) By (a), a nilpotent matrix can have no nonzero eigenvalues, i.e., all its eigenvalues are 0.
What is nilpotent matrix example?
A nilpotent matrix is a square matrix A such that Ak = 0. For a square matrix of order 2 x 2, to be a nilpotent matrix, the square of the matrix should be a null matrix, and for a square matrix of 3 x 3, to be a nilpotent matrix, the square or the cube of the matrix should be a null matrix.
What is the index of a nilpotent matrix?
The index of an n × n nilpotent matrix is always less than or equal to . For example, every 2 × 2 nilpotent matrix squares to zero. The determinant and trace of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible.
What is nilpotent index?
A nilpotent matrix (P) is a square matrix, if there exists a positive integer ‘m’ such that Pm = O. In other words, matrix P is called nilpotent of index m or class m if Pm = O and Pm-1 ≠ O. Here O is the null matrix (or zero matrix). Contents show. Nilpotent matrix Examples.
How do you find nilpotent?
If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent Let A be an n×n matrix such that tr(An)=0 for all n∈N. Then prove that A is a nilpotent matrix. Namely there exist a positive integer m such that Am is the zero matrix.
