minimal polynomial of

USAGE SUMMARY

The phrase "minimal polynomial of" is correct and usable in written English.
It is typically used in mathematics, particularly in linear algebra and abstract algebra, to refer to the unique monic polynomial of least degree that has a given linear transformation or matrix as a root. Example: "The minimal polynomial of the matrix A is x^2 - 5x + 6."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

22 human-written examples

Let and let be the minimal polynomial of Then (32).

Journal of Inequalities and Applications

Suppose that and let be the minimal polynomial of Assume that.

Journal of Inequalities and Applications

The minimal polynomial of T l is equal to the characteristic one.

Pacific Journal of Mathematics for Industry

In this paper, we determine the constant term of the minimal polynomial of λ q denoted by P q ∗ ( x ).

Fixed Point Theory and Applications

The minimal polynomial of λ q has been used for many aspects in the literature (see [5 8] and [9]).

Fixed Point Theory and Applications

Let f = Irr ( a, K ) be the minimal polynomial of a with respect to K and γ = w ( f ).

Fixed Point Theory and Applications

Human-verified similar examples from authoritative sources

Similar Expressions

38 human-written examples

The characteristic and the minimal polynomials of A are factorized as (z−2)2(z−3)6 and (z−2)2(z−3)3, respectively.

Pacific Journal of Mathematics for Industry

We also prove that the characteristic and the minimal polynomials of a constructed tridiagonal matrix are equal to each other.

Pacific Journal of Mathematics for Industry

The characteristic and the minimal polynomials of A and T are all the same polynomial with respect to z, which is factorized as (z−2+i 2 z−2−i 2 z−2)(z−2+i 2 z−2−i 2 z−2

Pacific Journal of Mathematics for Industry

The characteristic and the minimal polynomials of A and T are all the same polynomial with respect to z, which is factored as (z−2)6.

Pacific Journal of Mathematics for Industry

In [3], Cangul studied the minimal polynomials of the real part of ζ, i.e., of cos ( 2 π / n ) over the rationals.

Fixed Point Theory and Applications

Expert writing Tips

Best practice

When discussing algebraic elements, always specify the field over which you are considering the "minimal polynomial of". The field determines the coefficients allowed in the polynomial.

Common error

Avoid using "minimal polynomial of" interchangeably with the characteristic polynomial. While related, the characteristic polynomial may have a higher degree than the minimal polynomial.

Linguistic Context

The phrase "minimal polynomial of" functions as a mathematical term used to describe a specific polynomial associated with a matrix or an algebraic element. It identifies the monic polynomial of least degree that annihilates the given matrix or has the element as a root, as evidenced by Ludwig's examples.

Expression frequency: Common

Frequent in

Science

100%

Less common in

News & Media

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Formal & Business

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Academia

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Ludwig's WRAP-UP

In summary, the phrase "minimal polynomial of" is a key term in mathematics, specifically algebra and linear algebra, denoting the smallest degree polynomial that, when evaluated with a matrix or algebraic element, results in zero. Ludwig's analysis, consistent with its examples, indicates that this phrase is grammatically sound and frequently found in scientific and academic literature. When using the phrase, it's crucial to avoid confusion with the "characteristic polynomial of", and to specify the field under consideration. While primarily used in formal scientific contexts, understanding the "minimal polynomial of" provides foundational knowledge for more advanced algebraic studies.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

monic polynomial of least degree of

A formal restatement emphasizing both the monic property and the minimal degree.

smallest polynomial of

A more descriptive way to refer to the minimal polynomial, emphasizing its degree.

lowest degree polynomial of

Highlights that the polynomial has the lowest possible degree.

characteristic polynomial of

Focuses on the polynomial whose roots are the eigenvalues of the matrix, not necessarily the smallest degree.

annihilating polynomial of

Refers to any polynomial that, when applied to the matrix or element, results in zero, not specifically the minimal one.

irreducible polynomial of

Highlights the irreducibility of the polynomial over a specific field.

defining polynomial of

Emphasizes the polynomial that defines an algebraic extension.

field extension polynomial of

Connects the polynomial to the field extension it defines.

algebraic closure polynomial of

Relates the polynomial to the algebraic closure of a field.

root polynomial of

Highlights that the element is a root of the polynomial.

FAQs

What is the relationship between the characteristic polynomial and the "minimal polynomial of" a matrix?

The minimal polynomial divides the characteristic polynomial and they share the same roots, but the minimal polynomial is the polynomial of smallest degree for which the matrix evaluates to zero, while the characteristic polynomial's roots are the eigenvalues of the matrix.

How do you find the "minimal polynomial of" a matrix?

You can find the minimal polynomial by testing polynomials of increasing degree until you find one that annihilates the matrix (i.e., evaluates to the zero matrix). The "characteristic polynomial of" the matrix provides an upper bound for the degree you need to test.

What is the significance of the "minimal polynomial of" an algebraic element?

The minimal polynomial is the unique monic polynomial of smallest degree that has the algebraic element as a root. It plays a crucial role in understanding the structure of field extensions.

Can two different matrices have the same "minimal polynomial of"?

Yes, two different matrices can have the same minimal polynomial if they are similar, meaning they represent the same linear transformation under different bases. However, matrices with the same "characteristic polynomial of" may not have the same minimal polynomial.