# Cayley Hamilton Beispiel Essay

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 complexfield) satisfies its own characteristic equation.

If A is a given *n*×*n* matrix and *I _{n} * is the

*n*×

*n*identity matrix, then the characteristic polynomial of A is defined as

^{[7]}

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,

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 A^{0}, which is defined as the identity matrix. The theorem allows A^{n} 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^{[8]} in terms of inverses of linear functions of quaternions, a *non-commutative* ring, by Hamilton.^{[4]}^{[5]}^{[6]} This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. The theorem holds for general quaternionic matrices.^{[9]}^{[nb 1]} Cayley in 1858 stated it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case.^{[2]} The general case was first proved by Frobenius in 1878.^{[10]}

## Examples[edit]

### 1×1 matrices[edit]

For a 1×1 matrix *A* = (*a*_{1,1}), the characteristic polynomial is given by *p*(λ) = *λ* − *a*, and so *p*(*A*) = (*a*) − *a*_{1,1} = 0 is obvious.

### 2×2 matrices[edit]

As a concrete example, let

- .

Its characteristic polynomial is given by

The Cayley–Hamilton theorem claims that, if we *define*

then

We can verify by computation that indeed,

For a generic 2×2 matrix,

the characteristic polynomial is given by *p*(*λ*) = *λ*^{2} − (*a* + *d*)*λ* + (*ad* − *bc*), so the Cayley–Hamilton theorem states that

which is indeed always the case, evident by working out the entries of A^{2}.

## Applications[edit]

### Determinant and inverse matrix[edit]

See also: Determinant § Relation to eigenvalues and trace, and Characteristic polynomial § Properties

For a general *n*×*n*invertible matrixA, i.e., one with nonzero determinant, A^{−1} can thus be written as an (*n* − 1)-th order polynomial expression in A: As indicated, the Cayley–Hamilton theorem amounts to the identity

The coefficients *c*_{i} are given by the elementary symmetric polynomials of the eigenvalues of A. Using Newton identities, the elementary symmetric polynomials can in turn be expressed in terms of power sum symmetric polynomials of the eigenvalues:

where tr (*A*^{k}) is the trace of the matrix A^{k}. Thus, we can express *c*_{i} in terms of the trace of powers of A.

In general, the formula for the coefficients *c*_{i} is given in terms of complete exponential Bell polynomials as ^{[nb 2]}

In particular, the determinant of *A* corresponds to *c*_{0}. Thus, the determinant can be written as a trace identity

Likewise, the characteristic polynomial can be written as

and, by multiplying both sides by *A*^{−1} (note −(−1)^{n} = (−1)^{n−1}), one is led to an expression for the inverse of *A* as a trace identity,

For instance, the first few Bell polynomials are *B*_{0} = 1, *B*_{1}(*x*_{1}) = *x*_{1}, *B*_{2}(*x*_{1}, *x*_{2}) = *x*^{2}_{1} + *x*_{2}, and *B*_{3}(*x*_{1}, *x*_{2}, *x*_{3}) = *x*^{3}_{1} + 3 *x*_{1}*x*_{2} + *x*_{3}.

Using these to specify the coefficients *c _{i}* of the characteristic polynomial of a 2×2 matrix yields

The coefficient *c*_{0} gives the determinant of the 2×2 matrix, *c*_{1} minus its trace, while its inverse is given by

It is apparent from the general formula for *c _{n-k}*, expressed in terms of Bell polynomials, that this expression, ½((tr

*A*)

^{2}− tr(

*A*

^{2})), always gives the coefficient

*c*

_{n−2}of

*λ*

^{n−2}in the characteristic polynomial of any

*n*×

*n*matrix; so, for a 3×3 matrix A, the statement of the Cayley–Hamilton theorem can also be written as

where the right-hand side designates a 3×3 matrix with all entries reduced to zero. Likewise, this determinant in the *n* = 3 case, is now

This expression gives the negative of coefficient *c*_{n−3} of *λ*^{n−3} in the general case, as seen below.

Similarly, one can write for a 4×4 matrix A,

where, now, the determinant is *c*_{n−4},

and so on for larger matrices. The increasingly complex expressions for the coefficients *c*_{k} is deducible from Newton's identities or the Faddeev–LeVerrier algorithm.

Another method for obtaining these coefficients *c*_{k} for a general *n*×*n* matrix, provided no root be zero, relies on the following alternative expression for the determinant,

Hence, by virtue of the Mercator series,

where the exponential *only* needs be expanded to order *λ*^{−n}, since *p*(*λ*) is of order *n*, the net negative powers of *λ* automatically vanishing by the C–H theorem. (Again, this requires a ring containing the rational numbers.) The coefficients of λ can be directly written in terms of complete Bell polynomials by comparing this expression with the generating function of the Bell polynomial.

Differentiation of this expression with respect to λ allows determination of the generic coefficients of the characteristic polynomial for general n, as determinants of *m*×*m* matrices,^{[nb 3]}

^{[1]}Cayley proved the theorem for matrices of dimension 3 and less, publishing proof for the two-dimensional case.

^{[2]}

^{[3]}As for

*n*×

*n*matrices, Cayley stated “..., I have not thought it necessary to undertake the labor of a formal proof of the theorem in the general case of a matrix of any degree”.

^{[1]}

Hamilton proved that for a linear function of quaternions there exists a certain equation, depending on the linear function, that is satisfied by the linear function itself.

^{[4]}

^{[5]}

^{[6]}

where is the identity matrix. Cayley verified this identity for and 3 and postulated that it was true for all . For , direct verification gives

The Cayley-Hamilton theorem states that an matrix is annihilated by its characteristic polynomial, which is monic of degree .

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