Understanding Multiplicities

Question

I am having troubles understanding what 'multiplicities' mean. In example what does $-1/3(multiplicity 2)$ translate into??

To clarify

this is for finding zero's in a polynomial function

Any help would be appreciated!

Thanks

Answer 1

We want all polynomials of degree $n$ to have $n$ roots (or equivalently, $n$ linear factors).

But polynomials may have repeated roots. E.g. the polynomial $$f(x)=x^2-4x+4=(x-2)^2$$ has only one root, namely $2$. But we say it has multiplicity $2$ since its linear factor $(x-2)$ occurs twice in its factorization.

Another example: the polynomial $$x(x-2)^{100}(x+1)^2$$ has the root $0$ with multiplicity $1$, the root $2$ with multiplicity $100$ and the root $-1$ with multiplicity $2$.

Degree $n$ polynomials with real (or complex) coefficients have $n$ complex roots (counting multiplicities); this is the Fundamental Theorem of Arithmetic.

The main difficulty is factorizing the polynomial. As you may be aware, degree $\geq 5$ polynomials are not "solvable by radicals". But we can approximate their roots.

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My original answer (it's the same thing, but in a particular context): Most likely this refers to an eigenvalue of $-1/3$ with multiplicity $2$.

For example, the matrix $$A:=\begin{bmatrix} -1/3 & 2/3 \\ 0 & -1/3 \\ \end{bmatrix}$$ has the characteristic polynomial $$\det(A-\lambda I)=(\lambda+1/3)^2.$$ This means that $-1/3$ is an eigenvalue with multiplicity $2$ (as the exponent is $2$).

On the other hand, a matrix such as $$B:=\begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix}$$ has characteristic polynomial $$\det(B-\lambda I)=(\lambda-3)(\lambda+1)$$ so has eigenvalues $3$ and $-1$ each with multiplicity $1$.

If we count eigenvalues with their multiplicities, we ensure that an $n \times n$ matrix has $n$ eigenvalues.

Answer 2

Multiplicity usually means "the number of times it is repeated" (http://en.wikipedia.org/wiki/Multiplicity_%28mathematics%29), but without knowing what exactly your $-\frac{1}{3}$ is supposed to be, it's impossible to figure out what is repeated.