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characteristic and polynomial
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.
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.

characteristic and for
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.
The second involves something deeper, but its characteristic form focuses on a shift in policy for the community, not in the truth on which the community rests.
The three volumes brought to the fore a characteristic of Trevelyan's prose which remained conspicuous through his later works -- a genius for describing military action with clarity and with authority.
`` Young for his age '' means only the presence of some minor characteristic not quite usual.
Because of the falling characteristic of the rectifier, no ballast resistor was required for stability of operation.
For vector spaces over non-algebraically closed fields, we still need to find some substitute for characteristic values and vectors.
`` Increased boundary maintenance may be achieved, for example, by assigning a higher primacy or evaluation to activities characteristic of the external pattern.
Recently, for example, a paranoid woman's large-scale philosophizing, in the session, about the intrusive curiosity which has become, in her opinion, a deplorable characteristic of mid-twentieth-century human culture, developed itself, before the end of the session, into a suspicion that I was surreptitiously peeking at her partially exposed breast, as indeed I was.
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.
Now Rangoni comes to the point, and we hear, for the first time, a long, downward chromatic scale that will become the characteristic motif of his sinister power.
He has given only the one pass in his 27 innings, an unusual characteristic for a southpaw.
Although economic and personal circumstances vary widely among those now choosing apartments, Leo J. Pantas, vice president of a hardware manufacturing company, pointed out recently that many apartment seekers seem to have one characteristic in common: a desire for greater convenience and freedom from the problems involved in maintaining a house.
The organosulphur compounds responsible for the characteristic odour are believed to have antioxidant, antibiotic and anticarcinogenic properties, to stimulate the immune system and to be protective of liver functioning.
An ad hominem ( Latin for " to the man "), short for argumentum ad hominem, is an attempt to negate the truth of a claim by pointing out a negative characteristic or unrelated belief of the person supporting it .< ref >
As with those who engage other activities such as singing or running, the term may apply broadly to anyone who engages in it even briefly, or be more narrowly limited to those for whom it is a vocation, habit or characteristic practice.
The chromaffin cells of the medulla, named for their characteristic brown staining with chromic acid salts, are the body's main source of the circulating catecholamines epinephrine and norepinephrine.
One characteristic of the city is its painted gables, for which Aarau is sometimes called the " City of beautiful Gables ".
All elementary particles have a characteristic spin, for example electrons always have " spin 1 / 2 " while photons always have " spin 1 ".

characteristic and is
) 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.
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.
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.

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