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cayl
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There are 4 letters in CAYL ( A1C3L1Y4 )
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Definitions of cayl in various dictionaries:
CAYL - In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square m...
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In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation. * If A is a given n×n matrix and In is the n×n identity matrix, then the characteristic polynomial of A is defined as* * * * p * ( * λ * ) * = * det * ( * λ * * I * * n * * * − * A * ) * * , * * * {\displaystyle p(\lambda )=\det(\lambda I_{n}-A)~,} * where det is the determinant operation and λ is a scalar element of the base ring. Since the entries of the matrix are (linear or constant) polynomials in λ, the determinant is also an n-th order monic polynomial in λ. The Cayley–Hamilton theorem states that substituting the matrix A for λ in this polynomial results in the zero matrix, * * * * * p * ( * A * ) * = * O * . * * * {\displaystyle p(A)=O.} * The powers of A, obtained by substitution from powers of λ, are defined by repeated matrix multiplication; the constant term of p(λ) gives a multiple of the power A0, which is defined as the identity matrix. * The theorem allows An to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. * The theorem was first proved in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton. This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. The theorem holds for general quaternionic matrices. Cayley in 1858 stated it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case. The general case was first proved by Frobenius in 1878. |