Generalized Rayleigh entropyIt can be seen as an extension of Rayleigh entropy, which refers to the function R(A,B,x):

$latex {R{ \left( {A,B,x} \right) }\text{ }=\text{ }\frac{{x\mathop{{}}\nolimits^{{H}}Ax}}{{x\mathop{{}}\nolimits^{{H}}Bx}}}$latexR(A,B,x) = xHBxxHAx​

Where x is a non-zero vector, A and B are n×n Hermitan matrices, and B is a positive definite matrix. Let $latex {x\text{'}\text{ }=\text{ }B\mathop{{}}\nolimits^{{-1/2}}x}$latexx’ = B−1/2x , then the denominator can be transformed into:

$latex {x\mathop{{}}\nolimits^{{H}}Bx\text{ }=\text{ }x\text{'}\mathop{{}}\nolimits^{{H}}{ \left( {B\mathop{{}}\nolimits^{{-{{1}/{2}}}}} \right) }\mathop{{}}\nolimits^{{H}}BB\mathop{{}}\nolimits^{{-{{1}/{2}}}}x\text{'}\text{ }=\text{ }x\text{'}\mathop{{}}\nolimits^{{H}}B\mathop{{}}\nolimits^{{-{{1}/{2}}}}BB\mathop{{}}\nolimits^{{-{{1}/{2}}}}x\text{'}\text{ }={x\text{'}\mathop{{}}\nolimits^{{H}}x\text{'}}}$latexxHBx = x’H(B−1/2)HBB−1/2x’ = x’HB−1/2BB−1/2x’ =x’Hx’

The numerator is converted to:

$latex {x\mathop{{}}\nolimits^{{H}}Ax\text{ }=\text{ }x\text{'}\mathop{{}}\nolimits^{{H}}B\mathop{{}}\nolimits^{{-{{1}/{2}}}}AB\mathop{{}}\nolimits^{{-1/2}}x\text{'}}$latexxHAx = x’HB−1/2AB−1/2x’

at this time R(A,B,x) is transformed into ​R(A,B,x′) :

$latex {R{ \left( {A,B,x\text{'}} \right) }\text{ }=\text{ }\frac{{x\text{'}\mathop{{}}\nolimits^{{H}}B\mathop{{}}\nolimits^{{-1/2}}AB\mathop{{}}\nolimits^{{-1/2}}x\text{'}}}{{x\text{'}\mathop{{}}\nolimits^{{H}}x\text{'}}}}$latexR(A,B,x’) = x’Hx’x’HB−1/2AB−1/2x’​

From the above formula, we can conclude that generalized Rayleigh entropy can standardize the matrix and plays an important role in Fisher linear discriminant analysis.

References

【1】Rayleigh quotient and extreme value calculation

【2】Linear Discriminant Analysis LDA Principle Summary