What is the minimal polynomial of a nilpotent matrix?
If N is m-nilpotent, then its minimal polynomial is mN (x) = xm .
What is the meaning of minimal polynomial?
A monic polynomial is defined as a polynomial whose highest degree coefficient is equal to 1.
How do you show a matrix is nilpotent?
If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent Let A be an n×n matrix such that tr(An)=0 for all n∈N. Then prove that A is a nilpotent matrix. Namely there exist a positive integer m such that Am is the zero matrix.
What is minimal polynomial in field?
The minimal polynomial of an algebraic number is the unique irreducible monic polynomial of smallest degree with rational coefficients such that. and whose leading coefficient is 1.
How do you find the minimal polynomial?
The minimal polynomial is always well-defined and we have deg µA(X) ≤ n2. If we now replace A in this equation by the undeterminate X, we obtain a monic polynomial p(X) satisfying p(A) = 0 and the degree d of p is minimal by construction, hence p(X) = µA(X) by definition.
How do you find the nilpotent element?
An element x ∈ R , a ring, is called nilpotent if x m = 0 for some positive integer m. (1) Show that if n = a k b for some integers , then is nilpotent in . (2) If is an integer, show that the element a ― ∈ Z / ( n ) is nilpotent if and only if every prime divisor of also divides .
How do you prove something is nilpotent?
Theorem 1 If is a nilpotent matrix, then all its eigenvalues are zero. Conversely, if the eigenvalues of a square matrix are all zero, then is nilpotent. Clearly, if A q = 0 for some positive integer , then all eigenvalues of are zero; if has at least one eigenvalue which is nonzero, then A k ≠ 0 for all k ∈ Z ⩾ 0 .
How do you know if a polynomial is minimal?
The element α has a minimal polynomial when α is algebraic over F, that is, when f(α) = 0 for some non-zero polynomial f(x) in F[x]. Then the minimal polynomial of α is defined as the monic polynomial of least degree among all polynomials in F[x] having α as a root.
What is the difference between characteristic polynomial and minimal polynomial?
The characteristic polynomial of A is the product of all the elementary divisors. Hence, the sum of the degrees of the minimal polynomials equals the size of A. The minimal polynomial of A is the least common multiple of all the elementary divisors.
What is the minimal polynomial of a matrix?
In linear algebra, the minimal polynomial μ A of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μ A.
What is the characterization of a nilpotent matrix?
Characterization. The degree of an n × n nilpotent matrix is always less than or equal to n. For example, every 2 × 2 nilpotent matrix squares to zero. The determinant and trace of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible.
When do minimal and characteristic polynomials not factor according to their roots?
If the field F is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in F) alone, in other words they may have irreducible polynomial factors of degree greater than 1. For irreducible polynomials P one has similar equivalences: the kernel of P(A) has dimension at least 1.
What is the value of is nilpotent?
is nilpotent. . . is 0. The last theorem holds true for matrices over any field of characteristic 0 or sufficiently large characteristic. (cf. Newton’s identities ) . For example, every