Eigenvalue, eigenvector and eigenspace

Imagine you have a big building with lots of rooms. Each room may have different shapes and sizes. Some rooms are long and narrow, while others are wide and square. If you were to look at the whole building from above, you would see a shape that matches the overall size and shape of the building. You could call this the "eigenvalue" of the building - it represents the size and shape that the building as a whole has, no matter what the individual rooms look like.

Now, imagine you have a really cool flashlight. This flashlight shines a special light that, when it hits a wall, shows you which direction the wall is facing. You could shine this flashlight in every single room of the building and see which way each of the walls are facing. Some walls might face north, some might face south, and so on. If you were to write down all of the different directions that the walls are facing, you would end up with a list of "eigenvectors." Each eigenvector represents a direction that is important in helping to understand the overall shape and size of the building as a whole.

Finally, think about all of the walls that face in the same direction. If you were to gather up all of the walls that face north, for example, you would end up with a "space" that is made up entirely of walls that face in that direction. This is the "eigenspace" that corresponds to the eigenvector that represents the north direction. You could do the same thing for every other direction, and you would end up with a bunch of different eigenspaces, each made up of walls that face in the same important direction.

So basically, an eigenvalue is a way of describing the overall shape and size of something, like a building. Eigenvectors are directions that are important in understanding that overall shape and size. And eigenspaces are groups of things that all face in the same important direction.