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For example, every nilpotent matrix squares to zero. The determinant and trace of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible. The only nilpotent diagonalizable matrix is the zero matrix. See also: Jordan–Chevalley decomposition#Nilpotency criterion.
A matrix that has rank min(m, n) is said to have full rank; otherwise, the matrix is rank deficient. Only a zero matrix has rank zero. f is injective (or "one-to-one") if and only if A has rank n (in this case, we say that A has full column rank). f is surjective (or "onto") if and only if A has rank m (in this case, we say that A has full row ...
More generally, the trace of any idempotent matrix, i.e. one with A 2 = A, equals its own rank. The trace of a nilpotent matrix is zero. When the characteristic of the base field is zero, the converse also holds: if tr( A k ) = 0 for all k , then A is nilpotent.
Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with n = 2 {\displaystyle n=2} ). Both are linked, also through supersymmetry and Morse theory , [ 6 ] as shown by Edward Witten in a celebrated article.
The matrix exponential of another matrix (matrix-matrix exponential), [24] is defined as = = for any normal and non-singular n × n matrix X, and any complex n × n matrix Y. For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y , because the multiplication operator for ...
A matrix that preserves distances, i.e., a matrix that satisfies A * A = I where A * denotes the conjugate transpose of A. 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
There is a dual notion of co-rank of a finitely generated group G defined as the largest cardinality of X such that there exists an onto homomorphism G → F(X). Unlike rank, co-rank is always algorithmically computable for finitely presented groups, [17] using the algorithm of Makanin and Razborov for solving systems of equations in free groups.
Since L and M commute, the matrix L + M is nilpotent and I + (L + M)/2 is invertible with inverse given by a Neumann series. Hence L = M. If A is a matrix with positive eigenvalues and minimal polynomial p(t), then the Jordan decomposition into generalized eigenspaces of A can be deduced from the partial fraction expansion of p(t) −1.