The characteristic polynomial of a square matrix A is the determinant of the characteristic matrix xI - A.
Matrices + Characteristic Polynomial
, characteristic polynomial:
X - 4
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Evaluate
X-
=
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det=
X - 4
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The minimum polynomial of a square matrix A is the monic polynomial p(x) of smallest degree such that p(A) = 0. By the Cayley–Hamilton theorem , f (A) = 0 if f (x) is the characteristic polynomial of A. The minimum polynomial of A is a factor of the characteristic polynomial of A.
Matrices + Minimum Polynomial, Factor
, minimum polynomial: 16 - 8X + X2 =
X - 4
The minimum and characteristic polynomial operations have to return a variable for the polynomial. In the preceding examples, they returned X. However, the variable used depends on the matrix entries and you do not need to avoid X in the matrix. This point is illustrated in the following examples.
Matrices + Minimum Polynomial
, minimum polynomial: -5x + 3Xy +
-3X - y
λ + λ2
, characteristic polynomial: Θ2 -
y + λ
Θ + λy - xX