Eigenvalues of real symmetric matrices are always real numbers, and that's super exciting! Let's break down why this is true.

1. What Symmetry Means: A real symmetric matrix, which we can call $A$, has a special property: it looks the same when flipped over its diagonal. This is written as $A^T = A$. This symmetry is really important!

2. Finding Eigenvalues: We can find the eigenvalues using something called the characteristic polynomial. It's written as $p(\lambda) = \text{det}(A - \lambda I)$. This polynomial is made up of real numbers.

3. Understanding Roots: Real polynomials can have roots that are either real numbers or complex numbers. If they are complex, they come in pairs that are mirror images of each other. But for symmetric matrices, the roots, which are the eigenvalues, have to be real numbers!

4. Spectral Theorem: There’s a cool idea called the Spectral Theorem. It tells us that every real symmetric matrix can be changed into a diagonal matrix using something called an orthogonal matrix. This shows the special properties of the real eigenvalues.

So, let's celebrate the wonder of linear algebra! 🎉