Fixed-income attribution

2013-08-28T14:28:17Z

199.166.15.229: It is still not correct, one needs to distringuish between the ns rate and the ns discount (integrated) and not mix up the betas.

In [[Matrix (mathematics)|matrix theory]], '''Sylvester's formula''' or '''Sylvester's matrix theorem''' (named after [[James Joseph Sylvester|J. J. Sylvester]]) or '''Lagrange−Sylvester interpolation''' expresses an analytic [[matrix function|function]] ''f''(''A'') of a [[matrix (mathematics)|matrix]] ''A'' in terms of the [[eigenvalue, eigenvector and eigenspace|eigenvalues and eigenvectors]] of ''A''.<ref name=horn>
Roger A. Horn and Charles R. Johnson (1991), ''Topics in Matrix Analysis''. Cambridge University Press, ISBN 978-0-521-46713-1
</ref><ref name=claer>
Jon F. Claerbout (1976), ''Sylvester's matrix theorem'', a section of ''Fundamentals of Geophysical Data Processing''. [http://sepwww.stanford.edu/sep/prof/fgdp/c5/paper_html/node3.html Online version] at sepwww.stanford.edu, accessed on 2010-03-14.
</ref> It states that

::<math> f(A) = \sum_{i=1}^k f(\lambda_i) A_i ~,</math>
where the ''λ''''i'' are the eigenvalues of ''A'', and the matrices ''A''''i'' are the corresponding [[Frobenius covariant]]s of ''A'', matrix [[Lagrange polynomials]] of ''A''.

Sylvester's formula (1883) is only valid for [[diagonalizable matrix|diagonalizable matrices]]; an extension due to A. Buchheim (1886) covers the general case.

== Conditions ==
Sylvester's formula applies for any [[diagonalizable matrix]] ''A'' with ''k'' distinct eigenvalues, ''λ''1, …, ''λ''''k'', and any function ''f'' defined on some subset of the [[complex numbers]] such that ''f''(''A'') is well defined. The last condition means that every eigenvalue ''λ''''i'' is in the domain of ''f'', and that every eigenvalue ''λ''''i'' with multiplicity ''m''''i'' > 1 is in the interior of the domain, with ''f'' being (''m''''i'' − 1) times differentiable at ''λ''''i''.<ref name=horn/>{{rp|Def.6.4}}.

== Example ==

Consider the two-by-two matrix:
:<math> A = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}.</math>

This matrix has two eigenvalues, 5 and −2. Its Frobenius covariants are
:<math> \begin{align}
A_1 &= c_1 r_1 = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \begin{bmatrix} 1/7 & 1/7 \end{bmatrix} = \begin{bmatrix} 3/7 & 3/7 \\ 4/7 & 4/7 \end{bmatrix} = \frac{A+2I}{5-(-2)}\\
A_2 &= c_2 r_2 = \begin{bmatrix} 1/7 \\ -1/7 \end{bmatrix} \begin{bmatrix} 4 & -3 \end{bmatrix} = \begin{bmatrix} 4/7 & -3/7 \\ -4/7 & 3/7 \end{bmatrix}=\frac{A-5I}{-2-5}.
\end{align} </math>

Sylvester's formula then amounts to
:<math> f(A) = f(5) A_1 + f(-2) A_2. \, </math>

For instance, if {{mvar|f}} is defined by {{math|''f''(''x'') {{=}} ''x''−1}}, then Sylvester's formula expresses the matrix inverse {{math|''f''(''A'') {{=}} ''A''−1}} as
:<math> \frac{1}{5} \begin{bmatrix} 3/7 & 3/7 \\ 4/7 & 4/7 \end{bmatrix} - \frac{1}{2} \begin{bmatrix} 4/7 & -3/7 \\ -4/7 & 3/7 \end{bmatrix} = \begin{bmatrix} -0.2 & 0.3 \\ 0.4 & -0.1 \end{bmatrix}. </math>

==References==
{{reflist}}

[[Category:Matrix theory]]

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Fixed-income attribution