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Its Nature and Its Proof
Let A be an n×n matrix of real elements. The determinantal equation defining its eigenvalues is
where I is the n×n identity matrix. This equation in terms of a determinant generates a polynomimal equation p(λ)=0 where p(λ) is called the characteristic polynomial of the matrix. The Cayley-Hamilton Theorem is that if A is substituted for λ in the characteristic polynomial the result is a matrix of zeroes.
Let A be the matrix
Then the matrix A−λI is
The determinant of (A−λI) is then
This is the characteristic polynomial of the matrix A. The solutions to the polynomical equation
are the eigenvalues of the matrix A.
Consider now what results when λ is replaced with A and −2 is replaced with −2I.
A² is the matrix
Therefore −2I −5A + A² is
which evaluates to
Thus, almost mystically, A satisfies its own eigenvalue value equation.
The charactistic polynomial equation is equivalent to the determinantal equation
If A is substituted for Λ in this equation the result is
where O is a matrix of zeroes. Certainly A satisfies the determinantal equation.
Let the eigenvalue equation be expressed in the equivalent form
This removes the ambiguity of the sign of λn in the characteristic equation; it is always 1 for all n.
(To be continued.)
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