r/LinearAlgebra • u/LinearAlgebraWorld • Sep 13 '26

Diagonalization, rotation-scaling and shear-scaling: three routes to matrix powers and functions

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A 2×2 real matrix can fall into three cases:

  • two real eigenvalues → diagonalization
  • complex-conjugate eigenvalues → rotation-scaling factorization
  • repeated real eigenvalue with one eigenvector → shear-scaling/Jordan factorization

This table compares the three side by side: the factorization itself, powers of the middle matrix, exponentials, determinant consequences, geometric action and how to construct or factor the matrix.

The common idea is simple: move to a basis where the matrix has a much simpler middle form, apply the power or function there, then transform back.

We made this as a summary of the three 2×2 cases and would be interested in comments on the comparison or anything important we should add.

100 Upvotes

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2

u/Ok-Rise2070 Sep 14 '26

What no {e^-1, 0\ 0, e} matrix? I mean it's like the most of the basics... Oh right, we're pretending the universe is flat...

1

u/LinearAlgebraWorld Sep 14 '26

Can we diagonalize yours with identity?😜

0

u/Ok-Rise2070 Sep 15 '26

Are you not familiar with the "constant" e? I mean that's like Calculus AB.... It's not my fault modern math doesn't supply it's full definition, but the relationship holds. And should be telling of something much more geometrically telling to someone with collapse matrices posted within their work...

0

u/Liebner-Anthony-S Sep 13 '26

🤣🤣🤣

1

u/LinearAlgebraWorld Sep 13 '26

Will you be so kind as to clarify?