What Is Nilpotent Matrix?

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A nilpotent matrix is a square matrix with eigenvalues that are equal to zero. In general terms, this means that N ^ K = 0, where N is the square matrix, K is a positive integer (or whole number), and K is the degree of N.

Nilpotent matrices are used by mathematicians to explain what happens when a matrix or another type of equation equals zero when it is raised to a power. The word Nilpotent comes from the Latin root "potens," which means "possessing power," and "nil," which means "nothing."

Nilpotent matrices can be matrices that include zeros directly in the matrix, or they can be matrices that have no actual zeros in them. In fact, most nilpotent matrices have no zeros in the matrix itself. A matrix is nilpotent only if it squares to zero.

Consider a matrix with three columns and three rows as an example. The first row contains 5, 15 and 10. The second has -3, -9 and -6. The last row has 2, 6 and 4. Despite no zeros being present in the matrix, the matrix is still nilpotent, because these numbers square to zero. In addition, triangular matrices with zeros through main diagonals are also considered nilpotent.