What is the use of Frobenuis inner product?

Question

I was studying Linear Algebra from the famous textbook Linear Algebra by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence and reached the inner product of a vector space, I understand the intuition behind the standard dot product in $\mathbb{R}^{n}$, but he also mentioned Frobenuis inner product over the space of the square matrices. Is there any intuition I can understand, given that I only have the basic knowledge of algebra?