Determinant formulas
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A number of mathematical formulas can be written compactly using determinants. The following list contains some of the more useful or notable such formulas that have been discovered.
Extended quotient rule
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From the generalized product rule , if h =fg then
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{\displaystyle h'=fg'+f'g\,}
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{\displaystyle h''=fg''+2f'g'+f''g\,}
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{\displaystyle h'''=fg'''+3f'g''+3f''g'+f'''g\,}
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{\displaystyle \,\,\vdots }
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{\displaystyle h^{(n)}=fg^{(n)}+{n \choose 1}f'g^{(n-1)}+{n \choose 2}f''g^{(n-2)}+\dots +f^{(n)}g}
Using Cramer's rule to solve for f (n ) produces the determinant formula
[1]
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{\displaystyle f^{(n)}=\left({\frac {h}{g}}\right)^{(n)}=}
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{\displaystyle g^{-(n+1)}{\begin{vmatrix}g&&&&&h\\g'&g&&&&h'\\g''&2g'&g&&&h''\\g'''&3g''&3g'&g&&h'''\\\vdots &&&&\ddots &\\g^{(n)}&{n \choose 1}g^{(n-1)}&{n \choose 2}g^{(n-2)}&{n \choose 3}g^{(n-3)}&\cdots &h^{(n)}\end{vmatrix}}}
By applying this to find Taylor series coefficients in the cases h =x , g =ex -1; h =ex -1, g =ex +1; h =sin x , g =cos x ; h =x , g =sin x ; and h =1, g =cos x ; four different determinant expressions for the Bernoulli numbers and a determinant expression for the Euler numbers can be obtained.[2]
Symmetric polynomials
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The Schur polynomial
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{\displaystyle s_{(d_{1},d_{2},\dots ,d_{n})}(x_{1},x_{2},\dots ,x_{n})}
are defined as the quotients of the alternating polynomial
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{\displaystyle {\begin{vmatrix}x_{1}^{d_{1}+n-1}&x_{2}^{d_{1}+n-1}&\cdots &x_{n}^{d_{1}+n-1}\\x_{1}^{d_{2}+n-2}&x_{2}^{d_{2}+n-2}&\cdots &x_{n}^{d_{2}+n-2}\\\vdots &\vdots &\ddots &\vdots \\x_{1}^{d_{n}}&x_{2}^{d_{n}}&\cdots &x_{n}^{d_{n}}\end{vmatrix}}}
and the Vandermond determinant
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{\displaystyle {\begin{vmatrix}x_{1}^{n-1}&x_{2}^{n-1}&\cdots &x_{n}^{n-1}\\x_{1}^{n-2}&x_{2}^{n-2}&\cdots &x_{n}^{n-2}\\\vdots &\vdots &\ddots &\vdots \\1&1&\cdots &1\end{vmatrix}}}
This can, in turn, be expressed as a determinant involving the complete homogeneous symmetric polynomials as[3]
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{\displaystyle {\begin{vmatrix}h_{d_{1}+n-1}&h_{d_{1}+n-2}&\cdots &h_{d_{1}}\\h_{d_{2}+n-2}&h_{d_{2}+n-3}&\cdots &h_{d_{2}-1}\\\vdots &\vdots &\ddots &\vdots \\h_{d_{n}}&h_{d_{n}-1}&\cdots &h_{d_{n}-n+1}\\\end{vmatrix}}}
Newton's identities
A002135 Number of terms in a symmetric determinant (See Muir p. 112)
