Invariant subspace

An invariant subspace is like a special room inside a bigger room. When you are in the special room, you can't leave it and go to the bigger room. The special room always stays the same, no matter what happens in the bigger room.

In math, we use the idea of an invariant subspace to study matrices. A matrix is like a big math machine that can do all kinds of things to numbers. When we talk about the invariant subspace of a matrix, we mean a subset of the numbers that the matrix can't change. It's like the special room that the matrix can't get out of.

To figure out what an invariant subspace is for a matrix, we use some special rules. We look at how the matrix affects each point in the subspace, and we make sure that the result stays in the subspace. That's how we know it's invariant - it never changes no matter how many times we apply the matrix.

So, imagine you have a bunch of toys spread all over your room, but there is one special toy (let's call it a ball) that you can't touch or move. That ball is like the invariant subspace, it always stays in the same spot no matter how much you play in your room. Similarly, an invariant subspace of a matrix always stays the same, no matter how many times we apply the matrix.