orthogonal matrices preserve distances: O^T O = I means no stretching or squashing

How can you stretch or squash a square without changing its shape?

Imagine you have a square piece of paper. You want to fold it in a way that it still looks like a square but might look stretched or squashed.

Think of the square paper as a grid. If you rotate it perfectly, the grid lines stay straight and evenly spaced. This perfect rotation keeps the square's shape intact.

Example

Rotating a 2x2 square by 90 degrees keeps it a 2x2 square.

Remember this

Rotating an orthogonal matrix by 90 degrees doesn't change its shape, preserving distances.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

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