Diagonal transformations are really nice to visualize geometrically. In 2D they are just a combination of horizontal and vertical stretching.

While a generic matrix isn't quite this nice, if you can find a basis of eigenvectors, then the transformation "looks" like stretching and compressing along those eigenvectors by the values of the eigenvectors. This makes it pretty nice, but we can do better.

When we diagonalize a matrix, this is a composition of transformations. You first apply a changeofbasis to convert from the standard basis to the eigenbasis. Then you apply the nice diagonal transformation. Finally, you convert back.


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Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
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4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time a lot of time! sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

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This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

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