Given operators X and Y acting on a separable complex Hilbert space $mathcal{H}$, an interpolating operator is a bounded operator A such that AX=Y. In this article, we show the following: Let $Algmathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $mathcal{H}$ and let $X=(x_{ij});and;Y=(y_{ij})$ be operators in $mathcal{H}$. Then the following are equivalent: (1) There exists a normal operator $A=(a_{ij});in;Algmathcal{L}$ such that AX=Y. (2) There is a bounded sequence ${alpha_n};in;mathbb{C}$ such that $y_{ij}=alpha_jx_{ij};for;i,;j;{in};mathbb{N}$. Given vectors x and y in a separable complex Hilbert space $mathcal{H}$, an interpolating operator is a bounded operator A such that Ax=y. We show the following: Let $Algmathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $mathcal{H}$ and let $x=(x_i);and;y=(y_i)$ be vectors in $mathcal{H}$. Then the following are equivalent: (1) There exists a normal operator $A=(a_{ij});in;Algmathcal{L}$ such that Ax=y. (2) There is a bounded sequence ${alpha_n}$ in $mathbb{C}$ such that $y_i=alpha_ix_i;for;i{in}mathbb{N}$.