**Statement**

Suppose that *x*_{ij} for *i*,*j* = 1,...,*n* are commuting variables. Write *E*_{ij} 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 *E*_{ij} can be written in a matrix form:

where are matrices with elements *E*_{ij}, *x*_{ij}, 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

### Other articles related to "statement, statements":

... 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

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**statements**transfer control unconditionally ... There are four types of jump

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**statement**looks like this goto The identifier must be a label (followed by a colon) located in the current function ...

... 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 ...

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### Famous quotes containing the word statement:

“He that writes to himself writes to an eternal public. That *statement* only is fit to be made public, which you have come at in attempting to satisfy your own curiosity.”

—Ralph Waldo Emerson (1803–1882)

“Truth is used to vitalize a *statement* rather than devitalize it. Truth implies more than a simple *statement* of fact. “I don’t have any whisky,” may be a fact but it is not a truth.”

—William Burroughs (b. 1914)

“After the first powerful plain manifesto

The black *statement* of pistons, without more fuss

But gliding like a queen, she leaves the station.”

—Stephen Spender (1909–1995)