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Nilpotent matrices

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Definition of nilpotent matrix
Definition of index of nilpotency
Every nilpotent matrix is similar to a sum of nilpotent Jordan blocks
A complex square matrix is nilpotent iff all its eigenvalues are zero
Commuting diagonalizable and nilpotent decomposition over the complex numbers

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Commuting diagonalizable and nilpotent decomposition over the complex numbers

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Matrix diagonalizationClose match
Nilpotent matricesClose match
Diagonalizability requires splitting and equality of geometric and algebraic multiplicitiesClose match
Every nilpotent matrix is similar to a sum of nilpotent Jordan blocksClose match
Normal matricesClose match
Definition of nilpotent matrixClose match
An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n).Close match
An n-by-n matrix is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n.Close match
An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.Close match
Definition of diagonalizable matrixClose match
A complex matrix is unitarily diagonalizable iff it is normalClose match
A complex square matrix is nilpotent iff all its eigenvalues are zeroClose match
Every square matrix is similar to one in Jordan form.Close match
Definition of diagonal matrixRelated
Every complex square matrix is unitarily similar to an upper triangular matrixRelated
The minimal polynomial of a square matrix exists and is unique.Related
Definition of index of nilpotencyRelated
Equivalence theorem for nonsingular matrices: the determinant of A is nonzero.Related
Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.Related
Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.Related

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