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  2. Nilpotent matrix - Wikipedia

    en.wikipedia.org/wiki/Nilpotent_matrix

    For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix []. That is, if is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b 1, b 2 such that Nb 1 = 0 and Nb 2 = b 1. This classification theorem holds for matrices over any field. (It is not necessary for the field to be algebraically closed.)

  3. Jordan normal form - Wikipedia

    en.wikipedia.org/wiki/Jordan_normal_form

    The Jordan block corresponding to λ is of the form λI + N, where N is a nilpotent matrix defined as N ij = δ i,j−1 (where δ is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where f is a complex analytic function. For example, in principle the Jordan form could give a closed-form expression for the ...

  4. Nilpotent - Wikipedia

    en.wikipedia.org/wiki/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.

  5. Zero divisor - Wikipedia

    en.wikipedia.org/wiki/Zero_divisor

    A nilpotent element of a nonzero ring is always a two-sided zero divisor. An idempotent element of a ring is always a two-sided zero divisor, since () = = (). The ring of n × n matrices over a field has nonzero zero divisors if n ≥ 2.

  6. List of named matrices - Wikipedia

    en.wikipedia.org/wiki/List_of_named_matrices

    Laplacian matrix — a matrix equal to the degree matrix minus the adjacency matrix for a graph, used to find the number of spanning trees in the graph. Seidel adjacency matrix — a matrix similar to the usual adjacency matrix but with −1 for adjacency; +1 for nonadjacency; 0 on the diagonal. Skew-adjacency matrix — an adjacency matrix in ...

  7. Square root of a matrix - Wikipedia

    en.wikipedia.org/wiki/Square_root_of_a_matrix

    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.

  8. Matrix exponential - Wikipedia

    en.wikipedia.org/wiki/Matrix_exponential

    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 ...

  9. Companion matrix - Wikipedia

    en.wikipedia.org/wiki/Companion_matrix

    Not every square matrix is similar to a companion matrix, but every square matrix is similar to a block diagonal matrix made of companion matrices. If we also demand that the polynomial of each diagonal block divides the next one, they are uniquely determined by A , and this gives the rational canonical form of A .