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Determinant formulas

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

From the generalized product rule, if h=fg then

$h'=fg'+f'g\,$

$h''=fg''+2f'g'+f''g\,$

$h'''=fg'''+3f'g''+3f''g'+f'''g\,$

$\,\,\vdots$

$h^{(n)}=fg^{(n)}+{n \choose 1}f'g^{(n-1)}+{n \choose 2}f''g^{(n-2)}+\dots +f^{(n)}g$

Using Cramer's rule to solve for f⁽ⁿ⁾ produces the determinant formula

$f^{(n)}=\left({\frac {h}{g}}\right)^{(n)}=$

$g^{-(n+1)}{\begin{vmatrix}g&&&&&h\\g'&g&&&&h'\\g''&2g'&g&&&h''\\g'''&3g''&3g'&g&&h'''\\\vdots &&&&\ddots &\\g^{(n)}&{n \choose 1}g^{(n-1)}&{n \choose 2}g^{(n-2)}&{n \choose 3}g^{(n-3)}&\cdots &h^{(n)}\end{vmatrix}}$

By applying this to find Taylor series coefficients in the cases h=x, g=e^x-1; h=e^x-1, g=e^x+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.

Symmetric polynomials

The Schur polynomial

$s_{(d_{1},d_{2},\dots ,d_{n})}(x_{1},x_{2},\dots ,x_{n})$

are defined as the quotients of the alternating polynomial

${\begin{vmatrix}x_{1}^{d_{1}+n-1}&x_{2}^{d_{1}+n-1}&\cdots &x_{n}^{d_{1}+n-1}\\x_{1}^{d_{2}+n-2}&x_{2}^{d_{2}+n-2}&\cdots &x_{n}^{d_{2}+n-2}\\\vdots &\vdots &\ddots &\vdots \\x_{1}^{d_{n}}&x_{2}^{d_{n}}&\cdots &x_{n}^{d_{n}}\end{vmatrix}}$

and the Vandermond determinant

${\begin{vmatrix}x_{1}^{n-1}&x_{2}^{n-1}&\cdots &x_{n}^{n-1}\\x_{1}^{n-2}&x_{2}^{n-2}&\cdots &x_{n}^{n-2}\\\vdots &\vdots &\ddots &\vdots \\1&1&\cdots &1\end{vmatrix}}$

This can, in turn, be expressed as a determinant involving the complete homogeneous symmetric polynomials as

${\begin{vmatrix}h_{d_{1}+n-1}&h_{d_{1}+n-2}&\cdots &h_{d_{1}}\\h_{d_{2}+n-2}&h_{d_{2}+n-3}&\cdots &h_{d_{2}-1}\\\vdots &\vdots &\ddots &\vdots \\h_{d_{n}}&h_{d_{n}-1}&\cdots &h_{d_{n}-n+1}\\\end{vmatrix}}$

Newton's identities A002135 Number of terms in a symmetric determinant (See Muir p. 112)