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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}} | |||
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]]. | |||
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 | |||
| url = http://mathworld.wolfram.com/CharacteristicPolynomial.html | |||
| title = Characteristic Polynomial of a Graph - Wolfram MathWorld | |||
|accessdate = August 26, 2011}}</ref> | |||
==Motivation== | |||
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: | |||
:<math>(t-a_1)(t-a_2)(t-a_3)\cdots.\,</math> | |||
This works because the diagonal entries are also the eigenvalues of this matrix. | |||
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 | |||
:<math>A \mathbf{v} = \lambda \mathbf{v},\,</math> | |||
or | |||
:<math>(\lambda I - A)\mathbf{v} = 0\,</math> | |||
(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''. | |||
==Formal definition== | |||
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 | |||
:<math>p_A(t) = \det \left(t \boldsymbol{I} - A\right)</math> | |||
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''. | |||
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''). | |||
==Examples== | |||
Suppose we want to compute the characteristic polynomial of the matrix | |||
:<math>A=\begin{pmatrix} | |||
2 & 1\\ | |||
-1& 0 | |||
\end{pmatrix}. | |||
</math> | |||
We now compute the [[determinant]] of | |||
:<math>t I-A = \begin{pmatrix} | |||
t-2&-1\\ | |||
1&t-0 | |||
\end{pmatrix} | |||
</math> which is <math>(t-2)t - 1(-1) = t^2-2t+1 \,\!,</math> the characteristic polynomial of ''A''. | |||
Another example uses [[hyperbolic function]]s of a [[hyperbolic angle]] φ. | |||
For the matrix take | |||
:<math>A=\begin{pmatrix} \cosh(\phi) & \sinh(\phi)\\ \sinh(\phi)& \cosh(\phi) \end{pmatrix}.</math> | |||
Its characteristic polynomial is | |||
:<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> | |||
==Properties== | |||
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>) | |||
For a 2×2 matrix ''A'', the characteristic polynomial is thus given by | |||
: <math> t^2 - \operatorname{tr}(A) t + \operatorname{det}(A) </math>. | |||
Using the language of [[exterior algebra]], one may compactly express the characteristic polynomial of an ''n''×''n'' matrix ''A'' as | |||
: <math> p_A (t) = \sum_{k=0}^n t^{n-k} (-1)^k \operatorname{tr}(\Lambda^k A) </math> | |||
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, | |||
:<math>\frac{1}{k!} | |||
\begin{vmatrix} \operatorname{tr}A & k-1 &0&\cdots\\ | |||
\operatorname{tr}A^2 &\operatorname{tr}A& k-2 &\cdots\\ | |||
\cdots & \cdots & \cdots & \cdots \\ | |||
\operatorname{tr}A^{k-1} &\operatorname{tr}A^{k-2}& \cdots& 1 \\ | |||
\operatorname{tr}A^k &\operatorname{tr}A^{k-1}& \cdots& \operatorname{tr}A \\ \end{vmatrix} ~.</math> | |||
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''. | |||
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. | |||
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]]. | |||
==Characteristic polynomial of a product of two matrices== | |||
If ''A'' and ''B'' are two square ''n×n'' matrices then characteristic polynomials of ''AB'' and ''BA'' coincide: | |||
:<math>p_{AB}(t)=p_{BA}(t).\,</math> | |||
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. | |||
One has | |||
: <math> p_{BA}(t) = t^{n-m} p_{AB}(t).\,</math> | |||
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. | |||
When ''A'' is [[Non-singular_matrix|non-singular]] our result follows from the fact that ''AB'' and ''BA'' are [[similar matrices|similar]]: | |||
:<math>BA = A^{-1} (AB) A.\,</math> | |||
==Types== | |||
===Characteristic equation=== | |||
In [[linear algebra]], the ''characteristic equation'' (or ''secular equation'') of a square [[matrix (mathematics)|matrix]] ''A'' is the equation in one variable λ | |||
:<math>\det(A - \lambda I) = 0 \, </math> | |||
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]]. | |||
For example, the matrix | |||
:<math>A = \begin{bmatrix} 19 & 3 \\ -2 & 26 \end{bmatrix} </math> | |||
has the characteristic equation | |||
:<math>\begin{align} | |||
0 &{}= \det(A - \lambda I) \\ | |||
&{}= \det\begin{bmatrix} 19-\lambda & 3 \\ -2 & 26-\lambda \end{bmatrix} \\ | |||
&{}= 500-45\lambda+\lambda^2 \\ | |||
&{}= (25-\lambda)(20-\lambda) . | |||
\end{align}</math> | |||
The [[eigenvalue]]s of this matrix are therefore 20 and 25. | |||
Simple shortcuts exist for low dimension matrices. | |||
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 | |||
:<math>\det(A)-{\operatorname{tr}}(A)\lambda+\lambda^2,</math> | |||
with roots | |||
:<math>\lambda_{1,2} = \frac{\operatorname{tr}(A) \pm \sqrt{ \operatorname{tr}(A)^2 - 4\,\det(A)}}{2} .</math> | |||
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 | |||
:<math>\det(A)-c_2\lambda+{\operatorname{tr}}(A)\lambda^2-\lambda^3 ~.</math> | |||
Similarly, for a 4×4 matrix, it evaluates to | |||
:<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> | |||
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]]. | |||
The [[Cayley–Hamilton theorem]] states that every square matrix satisfies its own characteristic equation. | |||
===Secular function=== | |||
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. | |||
In [[linear algebra]], zeros of a secular function are the [[eigenvalues]] of a [[matrix (mathematics)|matrix]]. Characteristic polynomials also have eigenvalues as roots. | |||
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]]. | |||
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. | |||
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. | |||
===Secular equation=== | |||
''Secular equation'' has several meanings. | |||
In [[mathematics]] and [[numerical analysis]] it means characteristic equation. | |||
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 | |||
| url = http://dict.die.net/secular%20equation/ | |||
| title = secular equation | |||
|accessdate = January 21, 2010}}</ref> | |||
In [[molecular orbital]] calculations relating to the energy of the electron and its wave function it is also used instead of the characteristic equation. | |||
==See also== | |||
* [[Characteristic equation (disambiguation)|Characteristic equation]] | |||
* [[Invariants of tensors]] | |||
* [[Companion matrix]] | |||
==References== | |||
{{reflist}} | |||
* T.S. Blyth & E.F. Robertson (1998) ''Basic Linear Algebra'', p 149, Springer ISBN 3-540-76122-5 . | |||
* John B. Fraleigh & Raymond A. Beauregard (1990) ''Linear Algebra'' 2nd edition, p 246, [[Addison-Wesley]] ISBN 0-201-11949-8 . | |||
* Werner Greub (1974) ''Linear Algebra'' 4th edition, pp 120–5, Springer, ISBN 0-387-90110-8 . | |||
* Paul C. Shields (1980) ''Elementary Linear Algebra'' 3rd edition, p 274, [[Worth Publishers]] ISBN 0-87901-121-1 . | |||
* [[Gilbert Strang]] (1988) ''Linear Algebra and Its Applications'' 3rd edition, p 246, [[Brooks/Cole]] ISBN 0-15-551005-3 . | |||
==External links== | |||
* R. Skip Garibaldi. The characteristic polynomial and determinant are not ad hoc constructions. http://arxiv.org/abs/math/0203276 | |||
[[Category:Polynomials]] | |||
[[Category:Linear algebra]] | |||
[[Category:Tensors]] | |||
Revision as of 22:43, 18 April 2013
In linear algebra, every square matrix is associated with a characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace.
The characteristic polynomial of a 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.[1]
Motivation
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 a1, a2, a3, etc. then the characteristic polynomial will be:
This works because the diagonal entries are also the eigenvalues of this matrix.
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
or
(where I is the identity matrix). Since v is non-zero, this means that the matrix λ I − A is 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.
Formal definition
We start with a field K (such as the real or complex numbers) and an n×n matrix A over K. The characteristic polynomial of A, denoted by pA(t), is the polynomial defined by
where I denotes the n-by-n identity matrix and the determinant is being taken in K[t], the ring of polynomials in t over K.
Some authors define the characteristic polynomial to be det(A - t I). That polynomial differs from the one defined here by a sign (−1)n, 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).
Examples
Suppose we want to compute the characteristic polynomial of the matrix
We now compute the determinant of
Another example uses hyperbolic functions of a hyperbolic angle φ. For the matrix take
Its characteristic polynomial is
Properties
The polynomial pA(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 roots of pA(t) (this also holds for the minimal polynomial of A, but its degree may be less than n). The coefficients of the characteristic polynomial are all polynomial expressions in the entries of the matrix. In particular its constant coefficient pA (0) is det(−A) = (−1)n det(A), the coefficient of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
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Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
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For a 2×2 matrix A, the characteristic polynomial is thus given by
Using the language of exterior algebra, one may compactly express the characteristic polynomial of an n×n matrix A as
where tr(ΛkA) is the trace of the kth exterior power of A, with dimension , and may be evaluated explicitly as the determinant of the Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
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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 of A divides the characteristic polynomial of A.
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.
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.
Characteristic polynomial of a product of two matrices
If A and B are two square n×n matrices then characteristic polynomials of AB and BA coincide:
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. One has
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 2n2 + 1. When A is non-singular our result follows from the fact that AB and BA are similar:
Types
Characteristic equation
In linear algebra, the characteristic equation (or secular equation) of a square matrix A is the equation in one variable λ
where det is the determinant and I is the identity matrix. The solutions of the characteristic equation are precisely the eigenvalues 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.
For example, the matrix
has the characteristic equation
The eigenvalues of this matrix are therefore 20 and 25.
Simple shortcuts exist for low dimension matrices.
For a general 2×2 matrix A, the characteristic polynomial can be found from its determinant and trace, tr(A), to be
with roots
For a 3×3 matrix, c2= ½((trA)2−tr(A2)) is the sum of the principal minors of the matrix, and specifies the characteristic polynomial to be
Similarly, for a 4×4 matrix, it evaluates to
Expressions for n×n matrices are increasingly complicated, but tractable, cf. Newton's identities.
The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic equation.
Secular function
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 perturbations (on a time scale of a century, i.e. slow compared to annual motion) of planetary orbits, according to Lagrange's theory of oscillations.
In linear algebra, zeros of a secular function are the eigenvalues of a matrix. Characteristic polynomials also have eigenvalues as roots.
The characteristic polynomial is defined by the determinant of the matrix with a shift. It has zeros only, without any 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 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.
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.
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 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.
Secular equation
Secular equation has several meanings.
In mathematics and numerical analysis it means characteristic equation.
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.[4]
In molecular orbital calculations relating to the energy of the electron and its wave function it is also used instead of the characteristic equation.
See also
References
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- T.S. Blyth & E.F. Robertson (1998) Basic Linear Algebra, p 149, Springer ISBN 3-540-76122-5 .
- John B. Fraleigh & Raymond A. Beauregard (1990) Linear Algebra 2nd edition, p 246, Addison-Wesley ISBN 0-201-11949-8 .
- Werner Greub (1974) Linear Algebra 4th edition, pp 120–5, Springer, ISBN 0-387-90110-8 .
- Paul C. Shields (1980) Elementary Linear Algebra 3rd edition, p 274, Worth Publishers ISBN 0-87901-121-1 .
- Gilbert Strang (1988) Linear Algebra and Its Applications 3rd edition, p 246, Brooks/Cole ISBN 0-15-551005-3 .
External links
- R. Skip Garibaldi. The characteristic polynomial and determinant are not ad hoc constructions. http://arxiv.org/abs/math/0203276
- ↑ Template:Cite web
- ↑ Proposition 28 in these lecture notes
- ↑ Theorem 4 in these lecture notes
- ↑ Template:Cite web