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| {{Otheruses4|the characteristic polynomial of a matrix|the characteristic polynomial of a matroid|Matroid|that of a graded poset|Graded poset}}
| | A person that is obese is not able to function effectively.The bodily processes are moreover not carried correctly and therefore the person gets subjected to other diseases because well. With increasing weight, the immune program of the person begins to weaken and he is not able to fight the germs and bacteria attacking them.<br><br>The Oatmeal Diet. If you need to lose weight fast, consider following an oatmeal diet for 7 to 10 days. During this time period, oatmeal is the principal dish for all 3 food. Oatmeal is nutritious and high inside fiber. Eating oatmeal may aid you feel fuller plus decrease a appetite causing we to lose fat. Eat a little fresh fruit or plain steamed veggies as a side dish to aid keep the food balanced while following this diet.<br><br>The right time to do any cardio exercise is initially thing each morning before we eat anything. What happens is that because you haven't eaten anything throughout the evening, as you have been asleep, the body has to use calories that are stored in your body and burning calories is what we require if you would like to lose weight.<br><br>With all programs you must understand which you must be self disciplined and follow it properly. Here are some simple tricks for fat loss inside a hurry.<br><br>Everyone like to learn [http://safedietplansforwomen.com/how-to-lose-weight-fast lose weight fast] and you can lose it quickly utilizing pills however these have bad negative effects. It's also possible to starve oneself by going on 1 of the diets which cuts out food all together and to lose fat fast. You may have observed which most those who do lose weight this means put it all back on again.<br><br>For instance, I'm 25 years aged. 65% regarding my highest HR is 120 surpasses per minute (that is a pre-set calculation dependant on my age). Thus, based on the 65% beats per minute theory, We burn probably the many fat because I a rather long exercise having 120 surpasses per minute. In any event, I acquire issue with your pre-set max. Hour or thus being a similar for each time.<br><br>Drinking water enough to keep the body healthy will in turn assist in the optimal functioning of all body processes. It might confirm which digestion, intake plus excretion arises inside the best way in the body and might enable to remove all of the undigested plus waste from the body. Our body is filled with toxic components and removing them is 1 major task whenever struggling to get rid of fat quickly. With excessive water intake, it could aid inside flushing out these chemicals from the body plus thus keep the body clean. This refuses to mean that you have to keep drinking water alone, nevertheless include it well into the weight loss diet to get the maximum result from it. |
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| In [[linear algebra]], every [[square matrix]] is associated with a '''characteristic polynomial'''. This [[polynomial]] encodes several important properties of the [[matrix (mathematics)|matrix]], most notably its [[eigenvalue]]s, its [[determinant]] and its [[Trace (linear algebra)|trace]].
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| The '''characteristic polynomial of a [[graph (mathematics)|graph]]''' is the characteristic polynomial of its [[adjacency matrix]]. It is a [[graph invariant]], though it is not complete: the smallest pair of non-isomorphic graphs with the same characteristic polynomial have five nodes.<ref>{{cite web
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| | url = http://mathworld.wolfram.com/CharacteristicPolynomial.html
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| | title = Characteristic Polynomial of a Graph - Wolfram MathWorld
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| |accessdate = August 26, 2011}}</ref>
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| ==Motivation==
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| Given a square matrix ''A'', we want to find a polynomial whose zeros are the eigenvalues of ''A''. For a [[diagonal matrix]] ''A'', the characteristic polynomial is easy to define: if the diagonal entries are ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, etc. then the characteristic polynomial will be:
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| :<math>(t-a_1)(t-a_2)(t-a_3)\cdots.\,</math>
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| This works because the diagonal entries are also the eigenvalues of this matrix.
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| For a general matrix ''A'', one can proceed as follows. A scalar ''λ'' is an eigenvalue of ''A'' if and only if there is an [[eigenvector]] '''v''' ≠ 0 such that
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| :<math>A \mathbf{v} = \lambda \mathbf{v},\,</math>
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| or
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| :<math>(\lambda I - A)\mathbf{v} = 0\,</math>
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| (where '''''I''''' is the [[identity matrix]]). Since '''v''' is non-zero, this means that the matrix ''λ'' '''''I''''' − ''A'' is [[singular matrix|singular]] (non-invertible), which in turn means that its [[determinant]] is 0. Thus the roots of the function det(''λ'' '''''I''''' − ''A'') are the eigenvalues of ''A'', and it is clear that this determinant is a polynomial in ''λ''.
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| ==Formal definition==
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| We start with a [[field (mathematics)|field]] ''K'' (such as the [[real number|real]] or [[complex number|complex]] numbers) and an ''n''×''n'' matrix ''A'' over ''K''. The characteristic polynomial of ''A'', denoted by ''p''<sub>''A''</sub>(''t''), is the polynomial defined by
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| :<math>p_A(t) = \det \left(t \boldsymbol{I} - A\right)</math>
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| where '''''I''''' denotes the ''n''-by-''n'' [[identity matrix]] and the [[determinant]] is being taken in ''K''[''t''], the [[Polynomial ring|ring of polynomials]] in ''t'' over ''K''.
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| Some authors define the characteristic polynomial to be det(''A'' - ''t'' '''''I'''''). That polynomial differs from the one defined here by a sign (−1)<sup>''n''</sup>, so it makes no difference for properties like having as roots the eigenvalues of ''A''; however the current definition always gives a [[monic polynomial]], whereas the alternative definition always has constant term det(''A'').
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| ==Examples==
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| Suppose we want to compute the characteristic polynomial of the matrix
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| :<math>A=\begin{pmatrix}
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| 2 & 1\\
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| -1& 0
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| \end{pmatrix}.
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| </math> | |
| We now compute the [[determinant]] of
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| :<math>t I-A = \begin{pmatrix}
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| t-2&-1\\
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| 1&t-0
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| \end{pmatrix}
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| </math> which is <math>(t-2)t - 1(-1) = t^2-2t+1 \,\!,</math> the characteristic polynomial of ''A''.
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| Another example uses [[hyperbolic function]]s of a [[hyperbolic angle]] φ.
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| For the matrix take
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| :<math>A=\begin{pmatrix} \cosh(\phi) & \sinh(\phi)\\ \sinh(\phi)& \cosh(\phi) \end{pmatrix}.</math>
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| Its characteristic polynomial is
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| :<math>\det (tI - A) = (t - \cosh(\phi))^2 - \sinh^2(\phi) = t^2 - 2 t \ \cosh(\phi) + 1 = (t - e^\phi) (t - e^{-\phi}).</math>
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| ==Properties==
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| The polynomial ''p''<sub>''A''</sub>(''t'') is monic (its leading coefficient is 1) and its degree is ''n''. The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of ''A'' are precisely the [[root of a function|root]]s of ''p''<sub>''A''</sub>(''t'') (this also holds for the [[Minimal polynomial (linear algebra)|minimal polynomial]] of ''A'', but its degree may be less than ''n''). The coefficients of the characteristic polynomial are all [[polynomial expression]]s in the entries of the matrix. In particular its constant coefficient ''p''<sub>''A''</sub> (''0'') is det(−''A'') = (−1)<sup>''n''</sup> det(''A''), the coefficient of {{math|''t<sup>n</sup>''}} is one, and the coefficient of {{math|''t<sup>n−1</sup>''}} is tr(−''A'') = −tr(''A''), where {{math|tr(''A'')}} is the matrix [[trace (matrix)|trace]] of ''A''. (The signs given here correspond to the formal definition given in the previous section;<ref>Proposition 28 in these [http://users.math.yale.edu/~tl292/teaching/math225/notes/week10.pdf lecture notes]</ref> for the alternative definition these would instead be det(''A'') and (−1)<sup>''n'' − 1 </sup>tr(''A'') respectively.<ref>Theorem 4 in these [http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/week8.pdf lecture notes]</ref>)
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| For a 2×2 matrix ''A'', the characteristic polynomial is thus given by
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| : <math> t^2 - \operatorname{tr}(A) t + \operatorname{det}(A) </math>.
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| Using the language of [[exterior algebra]], one may compactly express the characteristic polynomial of an ''n''×''n'' matrix ''A'' as
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| : <math> p_A (t) = \sum_{k=0}^n t^{n-k} (-1)^k \operatorname{tr}(\Lambda^k A) </math>
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| where tr(''Λ<sup>k</sup>A)'' is the [[trace]] of the ''k<sup>th</sup>'' exterior power of ''A'', with dimension <math>\tbinom nk</math>, and may be evaluated explicitly as the determinant of the {{math|''k''×''k''}} matrix,
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| :<math>\frac{1}{k!}
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| \begin{vmatrix} \operatorname{tr}A & k-1 &0&\cdots\\
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| \operatorname{tr}A^2 &\operatorname{tr}A& k-2 &\cdots\\
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| \cdots & \cdots & \cdots & \cdots \\
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| \operatorname{tr}A^{k-1} &\operatorname{tr}A^{k-2}& \cdots& 1 \\
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| \operatorname{tr}A^k &\operatorname{tr}A^{k-1}& \cdots& \operatorname{tr}A \\ \end{vmatrix} ~.</math>
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| The [[Cayley–Hamilton theorem]] states that replacing ''t'' by ''A'' in the characteristic polynomial (interpreting the resulting powers as matrix powers, and the constant term ''c'' as ''c'' times the identity matrix) yields the zero matrix. Informally speaking, every matrix satisfies its own characteristic equation. This statement is equivalent to saying that the [[Minimal polynomial (linear algebra)|minimal polynomial]] of ''A'' divides the characteristic polynomial of ''A''.
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| Two [[similar matrices]] have the same characteristic polynomial. The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar.
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| The matrix ''A'' and its [[transpose]] have the same characteristic polynomial. ''A'' is similar to a [[triangular matrix]] [[if and only if]] its characteristic polynomial can be completely factored into linear factors over ''K'' (the same is true with the minimal polynomial instead of the characteristic polynomial). In this case ''A'' is similar to a matrix in [[Jordan normal form]].
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| ==Characteristic polynomial of a product of two matrices==
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| If ''A'' and ''B'' are two square ''n×n'' matrices then characteristic polynomials of ''AB'' and ''BA'' coincide:
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| :<math>p_{AB}(t)=p_{BA}(t).\,</math>
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| More generally, if ''A'' is ''m×n''-matrix and ''B'' is ''n×m'' matrices such that ''m''<''n'', then ''AB'' is ''m×m'' and ''BA'' is ''n×n'' matrix.
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| One has
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| : <math> p_{BA}(t) = t^{n-m} p_{AB}(t).\,</math>
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| To prove the first result, recognize that the equation to be proved, as a polynomial in t and in the entries of ''A'' and ''B'' is a universal polynomial identity. It therefore suffices to check it on an open set of parameter values in the complex numbers. The tuples (''A'',''B'',''t'') where ''A'' is an invertible complex ''n'' by ''n'' matrix, ''B'' is any complex ''n'' by ''n'' matrix, and ''t'' is any complex number from an open set in complex space of dimension 2''n''<sup>2</sup> + 1.
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| When ''A'' is [[Non-singular_matrix|non-singular]] our result follows from the fact that ''AB'' and ''BA'' are [[similar matrices|similar]]:
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| :<math>BA = A^{-1} (AB) A.\,</math>
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| ==Types==
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| ===Characteristic equation===
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| In [[linear algebra]], the ''characteristic equation'' (or ''secular equation'') of a square [[matrix (mathematics)|matrix]] ''A'' is the equation in one variable λ
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| :<math>\det(A - \lambda I) = 0 \, </math>
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| where det is the [[determinant]] and ''I'' is the [[identity matrix]]. The solutions of the characteristic equation are precisely the [[eigenvalue]]s of the matrix ''A''. The polynomial which results from evaluating the determinant is the characteristic polynomial of the matrix. The term "characteristic equation" is due to [[Wilhelm Killing]].
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| For example, the matrix
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| :<math>A = \begin{bmatrix} 19 & 3 \\ -2 & 26 \end{bmatrix} </math>
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| has the characteristic equation
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| :<math>\begin{align}
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| 0 &{}= \det(A - \lambda I) \\
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| &{}= \det\begin{bmatrix} 19-\lambda & 3 \\ -2 & 26-\lambda \end{bmatrix} \\
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| &{}= 500-45\lambda+\lambda^2 \\
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| &{}= (25-\lambda)(20-\lambda) .
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| \end{align}</math>
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| The [[eigenvalue]]s of this matrix are therefore 20 and 25.
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| Simple shortcuts exist for low dimension matrices.
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| For a general 2×2 matrix ''A'', the characteristic polynomial can be found from its [[determinant]] and [[trace (linear algebra)|trace]], tr(''A''), to be
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| :<math>\det(A)-{\operatorname{tr}}(A)\lambda+\lambda^2,</math>
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| with roots | |
| :<math>\lambda_{1,2} = \frac{\operatorname{tr}(A) \pm \sqrt{ \operatorname{tr}(A)^2 - 4\,\det(A)}}{2} .</math>
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| For a 3×3 matrix, ''c''<sub>2</sub>= ½((tr''A'')<sup>2</sup>−tr(''A''<sup>2</sup>)) is the sum of the [[principal minor]]s of the matrix, and specifies the characteristic polynomial to be
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| :<math>\det(A)-c_2\lambda+{\operatorname{tr}}(A)\lambda^2-\lambda^3 ~.</math>
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| Similarly, for a 4×4 matrix, it evaluates to
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| :<math>\lambda^4-(\mbox{tr}A)\lambda^3 + \frac{1}{2}\bigl((\mbox{tr}A)^2-\mbox{tr}(A^2)\bigr)\lambda^2 - \frac{1}{6}\bigl( (\mbox{tr}A)^3-3\mbox{tr}(A^2)(\mbox{tr}A)+2\mbox{tr}(A^3)\bigr)\lambda + \det(A) ~.</math>
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| Expressions for ''n''×''n'' matrices are increasingly complicated, but tractable, cf. [[Newton's identities#Expressing_elementary_symmetric_polynomials_in_terms_of_power_sums|Newton's identities]].
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| The [[Cayley–Hamilton theorem]] states that every square matrix satisfies its own characteristic equation.
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| ===Secular function===
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| The term ''secular function'' has also been used for what [[mathematicians]] now call a characteristic function of a linear operator (in some literature the term secular function is still used). The term comes from the fact that these functions were used to calculate [[secular phenomena|secular perturbations]] (on a time scale of a century, i.e. slow compared to annual motion) of planetary orbits, according to [[Joseph Louis Lagrange|Lagrange]]'s theory of oscillations.
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| In [[linear algebra]], zeros of a secular function are the [[eigenvalues]] of a [[matrix (mathematics)|matrix]]. Characteristic polynomials also have eigenvalues as roots.
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| The characteristic polynomial is defined by the [[determinant]] of the matrix with a shift. It has zeros only, without any [[Pole_(complex_analysis)|pole]]. Commonly, the secular function implies the characteristic polynomial. But, in the strict sense, the secular function has poles as well. Interestingly, the poles are located in the eigenvalues of its sub-matrices. Thus, if the information of the sub-matrices is available, the eigenvalues of the matrix can be described using that kind of information. Furthermore, by partitioning the matrix like matrix tearing or gruing, we can [[iterate]] the eigenvalues in a [[Recursion|recursive]] way. According to the methods of partitioning, the variant forms of the secular functions can be built up. However, they are all of the form of a series of the simple rational functions, which have poles at the eigenvalues of the partitioned matrices. For example, we can find a form of secular function in the [[divide-and-conquer eigenvalue algorithm]].
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| Recently, the secular function has been utilized in [[signal processing]]. The estimation problem with uncertainty involves a sort of eigenvalue problem, such as a bounded data uncertainty, [[total least squares]], data least squares, [[partial least squares]], [[errors-in-variables model]], etc. Many cases have been solved using their own secular equations. Some are still trying to find the unique secular equation that can resolve a given uncertainty estimation problem.
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| As for a numerical aspect, it is known that [[Newton's method]] is delicate when finding the roots of the secular equation. The higher-order interpolations are recommended. Among them, a [[simple rational approximation]] is a good choice considering the balance between the [[Stability theory|stability]] and the [[computational complexity]]. It is because the secular equation itself consists of a series of simple rational functions. However, using only interpolation cannot guarantee the stability. Thus fine search algorithms such as bisection steps are still required for accuracy.
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| ===Secular equation===
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| ''Secular equation'' has several meanings.
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| In [[mathematics]] and [[numerical analysis]] it means characteristic equation.
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| In [[astronomy]] it is the algebraic or numerical expression of the magnitude of the inequalities in a planet's motion that remain after the inequalities of a short period have been allowed for.<ref>{{cite web
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| | url = http://dict.die.net/secular%20equation/
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| | title = secular equation
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| |accessdate = January 21, 2010}}</ref>
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| In [[molecular orbital]] calculations relating to the energy of the electron and its wave function it is also used instead of the characteristic equation.
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| ==See also==
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| * [[Characteristic equation (disambiguation)|Characteristic equation]]
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| * [[Invariants of tensors]]
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| * [[Companion matrix]]
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| ==References==
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| {{reflist}}
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| * T.S. Blyth & E.F. Robertson (1998) ''Basic Linear Algebra'', p 149, Springer ISBN 3-540-76122-5 .
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| * John B. Fraleigh & Raymond A. Beauregard (1990) ''Linear Algebra'' 2nd edition, p 246, [[Addison-Wesley]] ISBN 0-201-11949-8 .
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| * Werner Greub (1974) ''Linear Algebra'' 4th edition, pp 120–5, Springer, ISBN 0-387-90110-8 .
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| * Paul C. Shields (1980) ''Elementary Linear Algebra'' 3rd edition, p 274, [[Worth Publishers]] ISBN 0-87901-121-1 .
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| * [[Gilbert Strang]] (1988) ''Linear Algebra and Its Applications'' 3rd edition, p 246, [[Brooks/Cole]] ISBN 0-15-551005-3 .
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| ==External links==
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| * R. Skip Garibaldi. The characteristic polynomial and determinant are not ad hoc constructions. http://arxiv.org/abs/math/0203276
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| [[Category:Polynomials]]
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| [[Category:Linear algebra]]
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| [[Category:Tensors]]
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A person that is obese is not able to function effectively.The bodily processes are moreover not carried correctly and therefore the person gets subjected to other diseases because well. With increasing weight, the immune program of the person begins to weaken and he is not able to fight the germs and bacteria attacking them.
The Oatmeal Diet. If you need to lose weight fast, consider following an oatmeal diet for 7 to 10 days. During this time period, oatmeal is the principal dish for all 3 food. Oatmeal is nutritious and high inside fiber. Eating oatmeal may aid you feel fuller plus decrease a appetite causing we to lose fat. Eat a little fresh fruit or plain steamed veggies as a side dish to aid keep the food balanced while following this diet.
The right time to do any cardio exercise is initially thing each morning before we eat anything. What happens is that because you haven't eaten anything throughout the evening, as you have been asleep, the body has to use calories that are stored in your body and burning calories is what we require if you would like to lose weight.
With all programs you must understand which you must be self disciplined and follow it properly. Here are some simple tricks for fat loss inside a hurry.
Everyone like to learn lose weight fast and you can lose it quickly utilizing pills however these have bad negative effects. It's also possible to starve oneself by going on 1 of the diets which cuts out food all together and to lose fat fast. You may have observed which most those who do lose weight this means put it all back on again.
For instance, I'm 25 years aged. 65% regarding my highest HR is 120 surpasses per minute (that is a pre-set calculation dependant on my age). Thus, based on the 65% beats per minute theory, We burn probably the many fat because I a rather long exercise having 120 surpasses per minute. In any event, I acquire issue with your pre-set max. Hour or thus being a similar for each time.
Drinking water enough to keep the body healthy will in turn assist in the optimal functioning of all body processes. It might confirm which digestion, intake plus excretion arises inside the best way in the body and might enable to remove all of the undigested plus waste from the body. Our body is filled with toxic components and removing them is 1 major task whenever struggling to get rid of fat quickly. With excessive water intake, it could aid inside flushing out these chemicals from the body plus thus keep the body clean. This refuses to mean that you have to keep drinking water alone, nevertheless include it well into the weight loss diet to get the maximum result from it.