Kronecker sum of discrete Laplacians
Okay, let's imagine that you have two sticker books. Each sticker book has a bunch of pages with empty spaces for stickers. Now let's say that each sticker represents a number. One sticker book is for your X numbers and the other is for your Y numbers.
The discrete Laplacian is like a special kind of sticker that helps you figure out how connected the numbers are. It looks at all the stickers in a page and checks how they're related to the other stickers. It then assigns a special value to each sticker based on how well it's connected.
Now, let's say you want to combine your two sticker books. You could just put all the pages together, but then you might have a hard time telling which stickers go with which numbers. So instead, you use something called the Kronecker sum.
The Kronecker sum is like taking each page in your X book and matching it with every page in your Y book. Then you put them together in a special way so that each sticker in your X book is in a new page with a sticker from your Y book.
Now, when you apply the discrete Laplacian to the Kronecker sum, it looks at each individual page and calculates how connected each sticker is to the stickers on the same page and on the adjacent pages. It then assigns a special value to each sticker based on how well it's connected in this new hybrid sticker book.
So, in summary, the Kronecker sum of discrete Laplacians is like combining two sets of numbers and figuring out how connected they are in a new, more complex system. It's like putting two sticker books together and analyzing how all the stickers relate to each other.
The discrete Laplacian is like a special kind of sticker that helps you figure out how connected the numbers are. It looks at all the stickers in a page and checks how they're related to the other stickers. It then assigns a special value to each sticker based on how well it's connected.
Now, let's say you want to combine your two sticker books. You could just put all the pages together, but then you might have a hard time telling which stickers go with which numbers. So instead, you use something called the Kronecker sum.
The Kronecker sum is like taking each page in your X book and matching it with every page in your Y book. Then you put them together in a special way so that each sticker in your X book is in a new page with a sticker from your Y book.
Now, when you apply the discrete Laplacian to the Kronecker sum, it looks at each individual page and calculates how connected each sticker is to the stickers on the same page and on the adjacent pages. It then assigns a special value to each sticker based on how well it's connected in this new hybrid sticker book.
So, in summary, the Kronecker sum of discrete Laplacians is like combining two sets of numbers and figuring out how connected they are in a new, more complex system. It's like putting two sticker books together and analyzing how all the stickers relate to each other.