We can do this through the characteristic values and vectors of T in certain special cases, i.e., when the minimal polynomial for T factors over the scalar field F into a product of distinct monic polynomials of degree 1.

Second, even if the characteristic polynomial factors completely over F into a product of polynomials of degree 1, there may not be enough characteristic vectors for T to span the space V.

The characteristic polynomial for A is Af and this is plainly also the minimal polynomial for A ( or for T ).

They are used to define the characteristic polynomial of a matrix that is an essential tool in eigenvalue problems in linear algebra.

The entries can be numbers or expressions ( as happens when the determinant is used to define a characteristic polynomial ); the definition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner.

That substitution yields the characteristic polynomial

More generally, if is the characteristic polynomial of a matrix A, then

By the fundamental theorem of algebra, applied to the characteristic polynomial of A, there is at least one eigenvalue λ < sub > 1 </ sub > and eigenvector e < sub > 1 </ sub >.

Another method based on the Cayley – Hamilton theorem finds an identity using the matrices ' characteristic polynomial, producing a more effective equation for A < sup > k </ sup > in which a scalar is raised to the required power, rather than an entire matrix.

It can be solved by methods described below yielding the closed-form expression which involve powers of the two roots of the characteristic polynomial t < sup > 2 </ sup > = t + 1 ; the generating function of the sequence is the rational function

The same coefficients yield the characteristic polynomial ( also " auxiliary polynomial ")

When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as ( x − r )< sup > 3 </ sup >, with the same root r occurring three times, then the solution would take the form

The characteristic polynomial equated to zero ( the characteristic equation ) is simply t − r = 0.

Solutions to such recurrence relations of higher order are found by systematic means, often using the fact that a < sub > n </ sub > = r < sup > n </ sup > is a solution for the recurrence exactly when t = r is a root of the characteristic polynomial.

Given a linearly recursive sequence, let C be the transpose of the companion matrix of its characteristic polynomial, that is

Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")

# Find the characteristic polynomial p ( t ).

the matrices satisfy various polynomials such as their minimal polynomials, which form a proper ideal ( because they are not all zero, in which case the result is trivial ); one might call this the characteristic ideal, by analogy with the characteristic polynomial.

Then the existence of an eigenvalue is equivalent to the ideal generated by ( the relations satisfied by ) being non-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix over an algebraically closed field by showing that the characteristic polynomial has a zero.

The eukaryotic cells possessed by all animals are surrounded by a characteristic extracellular matrix composed of collagen and elastic glycoproteins.

Note that the larger grains ( pebbles and gravel ) in the till are completely surrounded by the matrix of finer material ( silt and sand ), and this characteristic, known as matrix support, is diagnostic of till.

* There is a k-vector μ and a symmetric, nonnegative-definite k × k matrix Σ, such that the characteristic function of x is

In linear algebra, the Cayley – Hamilton theorem ( named after the mathematicians Arthur Cayley and William Hamilton ) states that every square matrix over a commutative ring ( such as the real or complex field ) satisfies its own characteristic equation.

If A is a given n × n matrix and I < sub > n </ sub > is the n × n identity matrix, then the characteristic polynomial of A is defined as

When the ring is a field, the Cayley – Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.

For a 1 × 1 matrix A = ( a ), the characteristic polynomial is given by p ( λ )= λ − a, and so p ( A )=( a )− a ( 1 )=( 0 ) is obvious.

For larger matrices, the expressions for the coefficients c < sub > k </ sub > of the characteristic polynomial in terms of the matrix components become increasingly complicated ; but they can also be expressed in terms of traces of powers of the matrix A, using Newton's identities ( at least when the ring contains the rational numbers ), thus resulting in more compact expressions ( but which involve divisions by certain integers ).

In fact, this expression, ½ (( trA )< sup > 2 </ sup >− tr ( A < sup > 2 </ sup >)), always gives the coefficient c < sub > n − 2 </ sub > of λ < sup > n − 2 </ sup > in the characteristic polynomial of any n × n matrix ; so, for a 3 × 3 matrix A, the statement of the Cayley – Hamilton theorem can also be written as

As the examples above show, obtaining the statement of the Cayley – Hamilton theorem for an n × n matrix requires two steps: first the coefficients c < sub > i </ sub > of the characteristic polynomial are determined by development as a polynomial in t of the determinant

Not all matrices are diagonalizable, but for matrices with complex coefficients many of them are: the set of diagonalizable complex square matrices of a given size is dense in the set of all such square matrices ( for a matrix to be diagonalizable it suffices for instance that its characteristic polynomial not have multiple roots ).

The matrix whose determinant is the characteristic polynomial of A is such a matrix, and since polynomials form a commutative ring, it has an adjugate

In linear algebra, every square matrix is associated with a characteristic polynomial.

The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.

) The concept of nationalism is the political principle that epitomizes and glorifies the territorial state as the characteristic type of socal structure.

Their rebellion against authoritarian society is not far removed from the violence of revolt characteristic of the juvenile delinquent.

It is worth dwelling in some detail on the crisis of this story, because it brings together a number of characteristic elements and makes of them a curious, riddling compound obscurely but centrally significant for Mann's work.

But it is characteristic of him, we are told, `` his little artifice '', to be able to introduce `` into a fairly vulgar and humorous piece of hackwork a sudden phrase of genuine creative art ''.

Mimesis is the nearest possible thing to the actual re-living of experience, in which the imagining person recovers through images something of the force and depth characteristic of experience itself.

A chief characteristic of experience in the mode of causal efficacy is one of derivation from the past.

A characteristic expression of such concern and inquiry is found in Joseph P. Lyford's Introduction To The Agreeable Autocracies, a recent paperback study of the institutions of modern democratic society.

Again, Henley's attitude of defiance which colors his ideal of self-mastery is far from characteristic of a Stoic thinker like Marcus Aurelius, whose gentle acquiescence is almost Christian, comparable to the patience expressed in Milton's sonnet on his own blindness.

It is a characteristic of thoughts that in re-thinking them we come, ipso facto, to understand why they were thought ''.

His nationalism was not a new characteristic, but its self-consciousness, even its self-satisfaction, is more obvious in a book that stretches over the long reach of English history.

A similar amateurish characteristic is revealed in Adams' failure to check the accuracy and authenticity of his informational sources.

The most obvious characteristic of contemporary American writing, apart from the beat nonsense, is its cosmopolitanism.

`` Do you suppose his self-consciousness is characteristic of the new Negro professionals or merely of doctors in general ''??

In snakes difference in size is a common characteristic of subspecies.

The process of boundary maintenance identifies and preserves the social system or subsystems, and the characteristic interaction is maintained.

It is no coincidence that the hebephrenic patient, the most severely dedifferentiated of all schizophrenic patients, shows, as one of his characteristic symptoms, laughter -- laughter which now makes one feel scorned or hated, which now makes one feel like weeping, or which now gives one a glimpse of the bleak and empty expanse of man's despair ; ;

Similarly, at the opposite end of the market cycle, towards the end of an intermediate or major decline, usually while the bottom is being formed on the price chart, it is characteristic that an increase is noticed in odd-lot selling again alerting the chartist that a bottom is becoming a greater likelihood.

In a society dominated by middle-class values and working in an institution which transmits and strengthens these social values, it is clear that the educational profession must work for the values which are characteristic of the society.

If one characteristic distinguishes Boris Godunov, it is the consistency with which every person on the stage -- including the chorus -- comes alive in the music.