- When an operator $\hat{P}$ operates on any function $\psi$ then this function $\psi$ changes into another function $\phi$. i.e.
$\hat{P} \psi =\phi$ Where $\phi$ is a new function linearly depending upon the initial function $\psi$.Example:Let us consider a function $f(x)=x^{2}$ and an operator ie. differential operator $\frac{d}{dx}$ is operate on the function. Then we get$\frac{d}{dx}f(x)= \frac{d}{dx} (x^{2})$$\frac{d}{dx}f(x)= 2x$Now the given function $f(x)=x^{2}$ change into another function $f(x)=x$. -
When an operator $\hat{P}$ operates on any function $\psi$ then this function $\psi$ does not change into another function but now this function $\psi$ may be with multiples of complex or real numbers(or values).i.e
$\hat{P} \psi =\lambda \phi$ Where $\lambda$ is Real OR Complex Number. This number or value is known as Eigenvalues.*In this case, the function $\psi$ is a member of the class of physically meaningful functions called the eigen function of the operator $\hat{P}$. The number $\lambda$ is called the eigen value of operator $\hat{P}$ associated with eigen function $\psi$ and this equation is known as the eigenvalue equation.*Example: Let us consider a function $f(x)=e^{2x}$ and an operator ie. differential operator $\frac{d}{dx}$ is operate on the function. Then we get$\frac{d}{dx}f(x)= \frac{d}{dx} (e^{2x})$$\frac{d}{dx}f(x)= 2e^{2x}$Now the given function $f(x)=e^{2x}$ change into another function $f(x)=x$.

A → Square matrix of $n$ order (where $n=1,2,3,.....$)

$\lambda$ → Scalar Factor

*Any value of $\lambda$ for which equation $(1)$ or equation $(2)$ has non zero (i.e $X \neq 0$) solution is called eigenvalues or characteristic roots or latent root of the matrix $A$ and corresponding non zero solution of $X$ is called eigenvectors or characteristic vectors or latent vectors of the matrix $A$*

*The matrix $\left| A- \lambda I \right|$ is called characteristic matrix of $A$. The determinant $\phi(\lambda) = \left| A- \lambda I \right|$ is called the characteristic polynomial of $A$*. So