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In linear algebra, a nilpotent matrix is a square matrix N such that = for some positive integer.The smallest such is called the index of , [1] sometimes the degree of .. More generally, a nilpotent transformation is a linear transformation of a vector space such that = for some positive integer (and thus, = for all ).
This definition can be applied in particular to square matrices.The matrix = is nilpotent because =.See nilpotent matrix for more.. In the factor ring /, the equivalence class of 3 is nilpotent because 3 2 is congruent to 0 modulo 9.
In the finite-dimensional case, 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 are on the superdiagonal [2] (this fact is used to prove the existence of Jordan canonical form). In turn this is equivalent to T n = 0 for some n. Therefore, for ...
If a 2 x 2 real matrix has zero trace, its square is a diagonal matrix. The trace of a 2 × 2 complex matrix is used to classify Möbius transformations. First, the matrix is normalized to make its determinant equal to one. Then, if the square of the trace is 4, the corresponding transformation is parabolic.
Nilpotent matrix: A square matrix satisfying A q = 0 for some positive integer q. Equivalently, the only eigenvalue of A is 0. Normal matrix: A square matrix that commutes with its conjugate transpose: AA ∗ = A ∗ A: They are the matrices to which the spectral theorem applies. Orthogonal matrix: A matrix whose inverse is equal to its ...
In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix. Let A be a square matrix. The index of A is the least nonnegative integer k such that rank(A k+1) = rank(A k). The Drazin inverse of A is the unique matrix A D that satisfies
Nilpotent matrices with complex entries form the main motivating case for the general theory, corresponding to the complex general linear group.From the Jordan normal form of matrices we know that each nilpotent matrix is conjugate to a unique matrix with Jordan blocks of sizes …, where is a partition of n.
Moreover, over a perfect field, a polynomial is separable if and only if it is square-free. Therefore an operator is potentially diagonalisable if and only if its minimal polynomial is square-free. In general (over any field), the minimal polynomial of a linear operator is square-free if and only if the operator is semisimple. [2] (In ...