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For a real $n \times n$ matrix $A$ the following are equivalent:

  • $A$ is symmetric and positive definite
  • There is an invertible matrix $P$ such that $A = P^{t} P$

This makes geometric sense when viewing $A$ as representing the dot product in some basis and the standard proof is based around this.

But is there a way to prove this viewing $A$ as a linear operator. What exactly is the geometric meaning of decomposing a symmetric positive definite operator into $P^{t}P$? What is the linear operator $P$ and $P^t$ doing that results in $P^t P =A$?

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