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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. Nilpotent - Wikipedia

    en.wikipedia.org/wiki/Nilpotent

    A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of a ring are precisely those that annihilate all integral domains internal to the ring (that is, of the form / for prime ideals ). This follows from the fact that nilradical is the intersection of all prime ...

  4. Matrix similarity - Wikipedia

    en.wikipedia.org/wiki/Matrix_similarity

    Similarity is an equivalence relation on the space of square matrices. Because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator: Rank; Characteristic polynomial, and attributes that can be derived from it:

  5. Engel's theorem - Wikipedia

    en.wikipedia.org/wiki/Engel's_theorem

    Note that if we merely have a Lie algebra of matrices which is nilpotent as a Lie algebra, then this conclusion does not follow (i.e. the naïve replacement in Lie's theorem of "solvable" with "nilpotent", and "upper triangular" with "strictly upper triangular", is false; this already fails for the one-dimensional Lie subalgebra of scalar ...

  6. Jordan normal form - Wikipedia

    en.wikipedia.org/wiki/Jordan_normal_form

    Then J 1 and J 2 are similar and have the same spectrum, including algebraic multiplicities of the eigenvalues. The procedure outlined in the previous paragraph can be used to determine the structure of these matrices. Since the rank of a matrix is preserved by similarity transformation, there is a bijection between the Jordan blocks of J 1 and ...

  7. Nilpotent operator - Wikipedia

    en.wikipedia.org/wiki/Nilpotent_operator

    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 ...

  8. Shift matrix - Wikipedia

    en.wikipedia.org/wiki/Shift_matrix

    Postmultiplication by a lower shift matrix results in a shift left. Similar operations involving an upper shift matrix result in the opposite shift. Clearly all finite-dimensional shift matrices are nilpotent; an n × n shift matrix S becomes the zero matrix when raised to the power of its dimension n. Shift matrices act on shift spaces.

  9. Nilpotent orbit - Wikipedia

    en.wikipedia.org/wiki/Nilpotent_orbit

    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.