A Memoir on the Theory of Matrices¶
Cayley, A. (1858). A Memoir on the Theory of Matrices. Philosophical Transactions of the Royal Society of London, 17-37.
Cited by¶
2 citations across 2 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Associativity
- (2) associativity is a property of a specific operation, independent of commutativity — many important operations are associative without being commutative, including function composition, matrix multiplication (Cayley 1858 established matrix algebra as an associative operation under multiplication)
This sourceFounds the theory of matrices
Supported in partVerified against the publisher's abstract
Cayley's abstract backs the non-commutativity half of the claim outright (matrices are 'not in general convertible') and treats matrices as single quantities that compound and take powers, but never states associativity in words.
“as regards their multiplication (or composition), there is the peculiarity that matrices are not in general convertible; it is nevertheless possible to form the powers (positive or negative, integral or fractional) of a matrix”
- (2) associativity is a property of a specific operation, independent of commutativity — many important operations are associative without being commutative, including function composition, matrix multiplication (Cayley 1858 established matrix algebra as an associative operation under multiplication)
- Commutativity
- … unavailable in non-commutative settings; (2) commutativity is a property of a specific operation on a specific structure, not a property of a set alone — addition and multiplication of integers are commutative, but subtraction and division are not; matrix multiplication is non-commutative (in general $AB \neq BA$)
This sourceFirst systematic paper on matrix algebra; establishes matrix multiplication as an associative but (in general) non-commutative operation — supports both the AB ≠ BA claim and the associative-non-commutative claim.
- … unavailable in non-commutative settings; (2) commutativity is a property of a specific operation on a specific structure, not a property of a set alone — addition and multiplication of integers are commutative, but subtraction and division are not; matrix multiplication is non-commutative (in general $AB \neq BA$)
Verification¶
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Registry ID ref:9fc17cef1745 · see in the full table