Suppose that xij for i,j = 1,...,n are commuting variables. Write Eij for the polarization operator
The Capelli identity states that the following differential operators, expressed as determinants, are equal:
Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded with all terms preserving their "left to right" order. Such a determinant is often called a column-determinant, since it can be obtained by the column expansion of the determinant starting from the first column. It can be formally written as
where in the product first come the elements from the first column, then from the second and so on. The determinant on the far right is Cayley's omega process, and the one on the left is the Capelli determinant.
The operators Eij can be written in a matrix form:
where are matrices with elements Eij, xij, respectively. If all elements in these matrices would be commutative then clearly . The Capelli identity shows that despite noncommutativity there exists a "quantization" of the formula above. The only price for the noncommutivity is a small correction: on the left hand side. For generic noncommutative matrices formulas like
do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices. That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof does not seem to exist. Direct verification of the statement can be given as an exercise for n' = 2, but is already long for n = 3.
Read more about this topic: Capelli's Identity
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... occurs at the beginning of the sentence and marks it as either a statement (bíi), a question (báa), et cetera in connected speech or writing, this particle is ... The evidence particle – this occurs at the end of statements and indicates the trustworthiness of the statement ... idiomatic translation bíi ril áya mahina wa statement present-tense beautiful/beautify flower observed-truth The flower is beautiful báa eril mesháad with question ...
... One mistake is "message-hunting" — looking only for a profitable statement or idea in a poem ... A short prose statement can work better than a poem for communicating advice ... attempt to mechanically combine the first two, defining poetry as the "beautiful statement of some high truth", or "truth" with "decorations" ...
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... if A requires B, it requires any true statement ... Thus, if at least one statement ought be true, every statement must materially entail it ought be true, and so every true statement ought be true ... if some statement ought be true then all statements that ought be true are true), consider the following logic ((U → !A) (A → ∩)) → (U → !∩) is a special case of axiom I, but its consequent ...
... C has two types of selection statements the if statement and the switch statement ... The if statement is in the form if () else In the if statement, if the in parentheses is nonzero (true), control passes to ... The switch statement causes control to be transferred to one of several statements depending on the value of an expression, which must have integral type ...
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