Dr Euler and characteristic equations

[Bob3141592]

In "Dr. Euler's Fabulous Formula" by Paul Nahin, early in chapter 1 is discussed characteristic polynomials of a square matrix and the Cayley-Hamilton theorem, that any square matrix A satisfies its own characteristic equation. On page 21 it states p(lambda) = lambda^2 + a1*lambda + a2 = 0 and then goes on to say "suppose we divide lambda^n by lambda^2 + a1*lambda + a2. The most general result is a polynomial of degree n - 2 and a remainder of at most degree one." Apparently this is an important result, but isn't this also an invalid divide by zero? Am I misunderstanding something here? I checked out characteristic equations in Schaum's Outline on Matrices but it doesn't touch on this point. Can somebody explain this? Thanks.

 

[matt grime]

That is not divide by zero. You're dividing a (definitely non-zero) polynomial into another. Remember to divide f by g in this sense means to find polys q and r with

f=qg+r

and deg(r)<deg(f).

 

[tacman]

The concept of characteristic equations and their application in mathematics can be quite complex and may require some clarification. It is understandable that you may have some confusion about the statement made in the book "Dr. Euler's Fabulous Formula" by Paul Nahin.

Firstly, let's define what a characteristic equation is. A characteristic equation is a polynomial equation that is derived from a square matrix. It is obtained by taking the determinant of the matrix and setting it equal to zero. The solutions of this equation are known as eigenvalues, and they have important applications in various fields of mathematics, such as linear algebra, differential equations, and physics.

In the statement you mentioned, p(lambda) = lambda^2 + a1*lambda + a2 = 0, lambda represents the eigenvalue of the matrix A, and a1 and a2 are coefficients of the characteristic equation. The Cayley-Hamilton theorem states that any square matrix A satisfies its own characteristic equation. This means that when we substitute the matrix A into the characteristic equation, the result will be equal to the zero matrix.

Now, let's come to the statement about dividing lambda^n by lambda^2 + a1*lambda + a2. This is not a division by zero. It is a polynomial division, where we are dividing a polynomial of degree n by a polynomial of degree 2. This results in a polynomial of degree n-2 and a remainder of at most degree one. This is a standard polynomial division process, similar to dividing numbers, where we obtain a quotient and a remainder.

The result of this division is not an eigenvalue or a solution to the characteristic equation. It is simply a polynomial expression that is obtained by dividing the characteristic equation by the eigenvalue. This process is used in solving systems of linear equations and finding the inverse of a matrix.

In conclusion, there is no invalid division by zero in the statement mentioned in the book. It is a mathematical process that is used to derive a polynomial expression from a characteristic equation. I hope this explanation helps to clarify your doubts. If you still have any further questions, please do not hesitate to ask.