Английская Википедия:Faddeev–LeVerrier algorithm

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Файл:Urbain Le Verrier.jpg

Urbain Le Verrier (1811–1877)
The discoverer of Neptune.

In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial <math>p_A(\lambda)=\det (\lambda I_n - A)</math> of a square matrix, Шаблон:Mvar, named after Dmitry Konstantinovich Faddeev and Urbain Le Verrier. Calculation of this polynomial yields the eigenvalues of Шаблон:Mvar as its roots; as a matrix polynomial in the matrix Шаблон:Mvar itself, it vanishes by the Cayley–Hamilton theorem. Computing the characteristic polynomial directly from the definition of the determinant is computationally cumbersome insofar as it introduces a new symbolic quantity <math>\lambda</math>; by contrast, the Faddeev-Le Verrier algorithm works directly with coefficients of matrix <math>A</math>.

The algorithm has been independently rediscovered several times in different forms. It was first published in 1840 by Urbain Le Verrier, subsequently redeveloped by P. Horst, Jean-Marie Souriau, in its present form here by Faddeev and Sominsky, and further by J. S. Frame, and others. (For historical points, see Householder. An elegant shortcut to the proof, bypassing Newton polynomials, was introduced by Hou. The bulk of the presentation here follows Gantmacher, p. 88.)

The Algorithm

The objective is to calculate the coefficients Шаблон:Math of the characteristic polynomial of the Шаблон:Math matrix Шаблон:Mvar,

<math>p_A(\lambda)\equiv \det(\lambda I_n-A)=\sum_{k=0}^{n} c_k \lambda^k~,</math>

where, evidently, Шаблон:Math = 1 and Шаблон:Math₀ = (−1)ⁿ det Шаблон:Mvar.

The coefficients Шаблон:Math are determined by induction on Шаблон:Mvar, using an auxiliary sequence of matrices

<math> \begin{align}

M_0 &\equiv 0 & c_n &= 1 \qquad &(k=0) \\ M_k &\equiv AM_{k-1} + c_{n-k+1} I \qquad \qquad & c_{n-k} &= -\frac 1 k \mathrm{tr}(AM_k) \qquad &k=1,\ldots ,n ~. \end{align}</math>

Thus,

<math>

M_1= I ~, \quad c_{n-1} = - \mathrm{tr} A =-c_n \mathrm{tr} A ; </math>

<math>M_2= A-I\mathrm{tr} A , \quad c_{n-2}=-\frac{1}{2}\Bigl(\mathrm{tr} A^2 -(\mathrm{tr} A)^2\Bigr) =

-\frac{1}{2} (c_n \mathrm{tr} A^2 +c_{n-1} \mathrm{tr} A) ; </math>

<math>M_3= A^2-A\mathrm{tr} A -\frac{1}{2}\Bigl(\mathrm{tr} A^2 -(\mathrm{tr} A)^2\Bigr) I,</math>

<math>c_{n-3}=- \tfrac{1}{6}\Bigl( (\operatorname{tr}A)^3-3\operatorname{tr}(A^2)(\operatorname{tr}A)+2\operatorname{tr}(A^3)\Bigr)=-\frac{1}{3}(c_n \mathrm{tr} A^3+c_{n-1} \mathrm{tr} A^2 +c_{n-2}\mathrm{tr} A); </math>

etc.,   ...;

<math>M_m= \sum_{k=1}^{m} c_{n-m+k} A^{k-1} ~,</math>

<math>c_{n-m}= -\frac{1}{m}(c_n \mathrm{tr} A^m+c_{n-1} \mathrm{tr} A^{m-1}+...+c_{n-m+1}\mathrm{tr} A)= -\frac{1}{m}\sum_{k=1}^{m} c_{n-m+k} \mathrm{tr} A^k ~  ; ...</math>

Observe Шаблон:Math terminates the recursion at Шаблон:Mvar. This could be used to obtain the inverse or the determinant of Шаблон:Mvar.

Derivation

The proof relies on the modes of the adjugate matrix, Шаблон:Math, the auxiliary matrices encountered.   This matrix is defined by

<math>(\lambda I-A) B = I ~ p_A(\lambda)</math>

and is thus proportional to the resolvent

<math>B = (\lambda I-A)^{-1} I ~ p_A(\lambda) ~. </math>

It is evidently a matrix polynomial in Шаблон:Math of degree Шаблон:Math. Thus,

<math>B\equiv \sum_{k=0}^{n-1} \lambda^k~ B_k= \sum_{k=0}^{n} \lambda^k~ M_{n-k} ,</math>

where one may define the harmless Шаблон:Math≡0.

Inserting the explicit polynomial forms into the defining equation for the adjugate, above,

<math>\sum_{k=0}^{n} \lambda^{k+1} M_{n-k} - \lambda^k (AM_{n-k} +c_k I)= 0~.</math>

Now, at the highest order, the first term vanishes by Шаблон:Math=0; whereas at the bottom order (constant in Шаблон:Mvar, from the defining equation of the adjugate, above),

<math>M_n A= B_0 A=c_0~,</math>

so that shifting the dummy indices of the first term yields

<math>\sum_{k=1}^{n} \lambda^{k} \Big ( M_{1+n-k} - AM_{n-k} +c_k I\Big )= 0~,</math>

which thus dictates the recursion

<math>\therefore \qquad M_{m} =A M_{m-1} +c_{n-m+1} I ~,</math>

for Шаблон:Mvar=1,...,Шаблон:Mvar. Note that ascending index amounts to descending in powers of Шаблон:Mvar, but the polynomial coefficients Шаблон:Mvar are yet to be determined in terms of the Шаблон:Mvars and Шаблон:Mvar.

This can be easiest achieved through the following auxiliary equation (Hou, 1998),

<math>\lambda \frac{\partial p_A(\lambda) }{ \partial \lambda} -n p =\operatorname{tr} AB~.</math>

This is but the trace of the defining equation for Шаблон:Mvar by dint of Jacobi's formula,

<math>\frac{\partial p_A(\lambda)}{\partial \lambda}= p_A(\lambda) \sum^\infty _{m=0}\lambda ^{-(m+1)} \operatorname{tr}A^m =

p_A(\lambda) ~ \operatorname{tr} \frac{I}{\lambda I -A}\equiv\operatorname{tr} B~. </math>

Inserting the polynomial mode forms in this auxiliary equation yields

<math>\sum_{k=1}^{n} \lambda^{k} \Big ( k c_k - n c_k - \operatorname{tr}A M_{n-k}\Big )= 0~,</math>

so that

<math>\sum_{m=1}^{n-1} \lambda^{n-m} \Big ( m c_{n-m} + \operatorname{tr}A M_{m}\Big )= 0~,</math>

and finally

<math>\therefore \qquad c_{n-m} = -\frac{1}{m} \operatorname{tr}A M_{m} ~.</math>

This completes the recursion of the previous section, unfolding in descending powers of Шаблон:Mvar.

Further note in the algorithm that, more directly,

<math> M_{m} =A M_{m-1} - \frac{1}{m-1} (\operatorname{tr}A M_{m-1}) I ~,</math>

and, in comportance with the Cayley–Hamilton theorem,

<math> \operatorname{adj}(A) =(-1)^{n-1} M_{n}=(-1)^{n-1} (A^{n-1}+c_{n-1}A^{n-2}+ ...+c_2 A+ c_1 I)=(-1)^{n-1} \sum_{k=1}^n c_k A^{k-1}~.</math>

The final solution might be more conveniently expressed in terms of complete exponential Bell polynomials as

<math> c_{n-k} = \frac{(-1)^{n-k}}{k!} {\mathcal B}_k \Bigl ( \operatorname{tr}A , -1! ~ \operatorname{tr}A^2, 2! ~\operatorname{tr}A^3, \ldots, (-1)^{k-1}(k-1)! ~ \operatorname{tr}A^k\Bigr ) .</math>

Example

<math>{\displaystyle A=\left[{\begin{array}{rrr}3&1&5\\3&3&1\\4&6&4\end{array}}\right]}</math>

<math>{\displaystyle {\begin{aligned}M_{0}&=\left[{\begin{array}{rrr}0&0&0\\0&0&0\\0&0&0\end{array}}\right]\quad &&&c_{3}&&&&&=&1\\M_{\mathbf {\color {blue}1} }&=\left[{\begin{array}{rrr}1&0&0\\0&1&0\\0&0&1\end{array}}\right] &A~M_{1}&=\left[{\begin{array}{rrr}\mathbf {\color {red}3} &1&5\\3&\mathbf {\color {red}3} &1\\4&6&\mathbf {\color {red}4} \end{array}}\right] &c_{2}&&&=-{\frac {1}{\mathbf {\color {blue}1} }} \mathbf {\color {red}10} &&=&-10\\M_{\mathbf {\color {blue}2} }&=\left[{\begin{array}{rrr}-7&1&5\\3&-7&1\\4&6&-6\end{array}}\right]\qquad &A~M_{2}&=\left[{\begin{array}{rrr}\mathbf {\color {red}2} &26&-14\\-8&\mathbf {\color {red}-12} &12\\6&-14&\mathbf {\color {red}2} \end{array}}\right]\qquad &c_{1}&&&=-{\frac {1}{\mathbf {\color {blue}2} }} \mathbf {\color {red}(-8)} &&=&4\\M_{\mathbf {\color {blue}3} }&=\left[{\begin{array}{rrr}6&26&-14\\-8&-8&12\\6&-14&6\end{array}}\right]\qquad &A~M_{3}&=\left[{\begin{array}{rrr}\mathbf {\color {red}40} &0&0\\0&\mathbf {\color {red}40} &0\\0&0&\mathbf {\color {red}40} \end{array}}\right]\qquad &c_{0}&&&=-{\frac {1}{\mathbf {\color {blue}3} }}\mathbf {\color {red}120} &&=&-40\end{aligned}}} </math>

Furthermore, <math>{\displaystyle M_{4}=A~M_{3}+c_{0}~I=0}</math>, which confirms the above calculations.

The characteristic polynomial of matrix Шаблон:Mvar is thus <math>{\displaystyle p_{A}(\lambda )=\lambda ^{3}-10\lambda ^{2}+4\lambda -40}</math>; the determinant of Шаблон:Mvar is <math>{\displaystyle \det(A)=(-1)^{3} c_{0}=40}</math>; the trace is 10=−c₂; and the inverse of Шаблон:Mvar is

<math>{\displaystyle A^{-1}=-{\frac {1}{c_{0}}}~ M_{3}={\frac {1}{40}} \left[{\begin{array}{rrr}6&26&-14\\-8&-8&12\\6&-14&6\end{array}}\right]=\left[{\begin{array}{rrr}0{.}15&0{.}65&-0{.}35\\-0{.}20&-0{.}20&0{.}30\\0{.}15&-0{.}35&0{.}15\end{array}}\right]} </math>.

An equivalent but distinct expression

A compact determinant of an Шаблон:Mvar×Шаблон:Mvar-matrix solution for the above Jacobi's formula may alternatively determine the coefficients Шаблон:Mvar,

<math>c_{n-m} = \frac{(-1)^m}{m!}

\begin{vmatrix} \operatorname{tr}A & m-1 &0&\cdots&0\\ \operatorname{tr}A^2 &\operatorname{tr}A& m-2 &\cdots&0\\

\vdots & \vdots & & & \vdots    \\

\operatorname{tr}A^{m-1} &\operatorname{tr}A^{m-2}& \cdots & \cdots & 1 \\ \operatorname{tr}A^m &\operatorname{tr}A^{m-1}& \cdots & \cdots & \operatorname{tr}A \end{vmatrix} ~.</math>

See also

• Characteristic polynomial
• Exterior algebra § Leverrier's algorithm
• Horner's method
• Fredholm determinant

References

Шаблон:Reflist Barbaresco F. (2019) Souriau Exponential Map Algorithm for Machine Learning on Matrix Lie Groups. In: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science, vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_10