# Capelli's Identity - Generalizations - The Mukhin–Tarasov–Varchenko Identity and The Gaudin Model - Statement

Statement

Consider matrices X and D as in Capelli's identity, i.e. with elements and at position (ij).

Let z be another formal variable (commuting with x). Let A and B be some matrices which elements are complex numbers.

$detleft( frac{partial}{partial_z} - A - X frac{1}{z-B} D^t right)$
$={det}^text{calculate as if all commute}_{text{Put all }xtext{ and }ztext{ on the left, while all derivations on the right}}$
$left( frac{partial}{partial_z} - A - X frac{1}{z-B} D^t right)$

Here the first determinant is understood (as always) as column-determinant of a matrix with non-commutative entries. The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left, while derivations on the right. (Such recipe is called a Wick ordering in the quantum mechanics).

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### Famous quotes containing the word statement:

Children should know there are limits to family finances or they will confuse “we can’t afford that” with “they don’t want me to have it.” The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.
Jean Ross Peterson (20th century)

If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.
—J.L. (John Langshaw)

The parent is the strongest statement that the child hears regarding what it means to be alive and real. More than what we say or do, the way we are expresses what we think it means to be alive. So the articulate parent is less a telling than a listening individual.
Polly Berrien Berends (20th century)