Also all such eigenvalues are simple roots of the characteristic polynomial.

Find the roots of the characteristic polynomial of 'A'.

Then he studied the roots of the characteristic polynomial , where .

The eigenvalues of a matrix can be determined by finding the roots of the characteristic polynomial.

Since the eigenvalues are roots of the characteristic polynomial, an matrix has at most eigenvalues.

Thus eigenvalue algorithms that work by finding the roots of the characteristic polynomial can be ill-conditioned even when the problem is not.

However, finding the roots of the characteristic polynomial may be an Condition number problem even when the underlying eigenvalue problem is well-conditioned.

Combining the two claims above reveals that the Perron-Frobenius eigenvalue "r" is simple root of the characteristic polynomial.

For stationarity, nonlinear inequality constraints can be placed to force roots of the characteristic polynomials to be outside the unit circle.

First, it requires finding all eigenvalues, say as roots of the characteristic polynomial, but it may not be possible to give an explicit expression for them.

This section proves that the Perron-Frobenius eigenvalue is a simple root of the characteristic polynomial of the matrix.

In other words, the roots of the characteristic polynomial corresponding to the matrix are not necessarily computed ahead in order to obtain its Schur decomposition.

This character has the property that for a prime ideal p of good reduction, the value χ(p) is a root of the characteristic polynomial of the Frobenius endomorphism.

In particular, the sum of the x, which is the k-th power sum s of the roots of the characteristic polynomial of A, is given by its trace:

The algebraic multiplicity of is its multiplicity as a root of the characteristic polynomial, that is, the largest integer such that divides evenly that polynomial.

This usage carries over to eigenproblems: a degenerate eigenvalue (i.e. a multiple coinciding root of the characteristic polynomial) is one that has more than one linearly independent eigenvector.

The Perron-Frobenius eigenvalue is simple: r is a simple root of the characteristic polynomial of A. Consequently, the eigenspace associated to r is one-dimensional.

If the roots of the characteristic polynomial ρ all have modulus less than or equal to 1 and the roots of modulus 1 are of multiplicity 1, we say that the root condition is satisfied.

The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts.

The factorization procedure is purely algebraic, the number of possible factirzations depends on the number of simple roots of the Characteristic polynomial (also called symbol) of the initial LPDO and reduced LPDOs appearing at each factorization step.

There does exist an algorithm to test whether a linear recurrence has infinitely many zeros, and if so to construct a decomposition of the positions of those zeros into periodic subsequences, based on the algebraic properties of the roots of the characteristic polynomial of the given recurrence.

In this case one can check that the index of for is equal to its multiplicity as a root of the minimal polynomial of (whereas, by definition, its algebraic multiplicity for , , is its multiplicity as a root of the characteristic polynomial of , i.e. ).