Newton's identities and matrices

Thread starter Jhenrique
Start date May 21, 2014
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identities Matrices

In summary, Newton's identities are a set of mathematical equations that relate the coefficients of a polynomial to its roots. They can be used to calculate the coefficients of a polynomial by using the eigenvalues and eigenvectors of its associated matrix, known as the Cayley-Hamilton theorem. They are important in algebra, number theory, physics, and engineering. However, they only work for polynomials with distinct roots and can become more complex and time-consuming for polynomials with a large number of terms.

May 21, 2014

Jhenrique

About the Newton's identities:

I'm right if I state that e_k = I_k, p_k = tr(A^k) and h_k = det(A^k) (being I_k the kth-invariant of the matrix A)?

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May 24, 2014

Jhenrique

PS: being ##A = \begin{bmatrix}
x_1 & 0 & \cdots & 0 \\
0 & x_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & x_n \\
\end{bmatrix}##

Related to Newton's identities and matrices

1. What are Newton's identities?

Newton's identities are a set of mathematical equations that relate the coefficients of a polynomial to its roots. They are named after Sir Isaac Newton, who first discovered them.

2. How do Newton's identities relate to matrices?

Newton's identities can be used to calculate the coefficients of a polynomial by using the eigenvalues and eigenvectors of its associated matrix. This is known as the Cayley-Hamilton theorem.

3. What is the significance of Newton's identities in mathematics?

Newton's identities are important in algebra and number theory, as they provide a way to find the coefficients of a polynomial without having to explicitly solve for its roots. They also have applications in physics and engineering.

4. Can Newton's identities be used for polynomials with complex roots?

Yes, Newton's identities can be used for polynomials with complex roots. However, the calculations may become more complex as the number of complex roots increases.

5. Are there any limitations to using Newton's identities?

One limitation of Newton's identities is that they only work for polynomials with distinct roots. They cannot be used for polynomials with repeated roots. Additionally, the calculations can become tedious and time-consuming for polynomials with a large number of terms.