In my opinion, "such that" is just like the word "so". But in the sentence below, substituting the phrase "such that" using "so" makes me confused.

"Let P and Q be m × n and m × l matrices" can not guarantee "PTP is nonsingular" after all. So, how should I understand the phrase "such that" in these sentence?

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In the context of a mathematical definition, "such that" is a more specific version of "so". In this example:

  • Q has been defined to be any m × l matrix.
  • P has been defined to be an m × n matrix.
  • P is restricted in some way.
  • We can conclude from the restriction on P that PTP is nonsingular. In other words, "so".

Notice that the restriction on P is not explicitly stated. Instead, it is just enough of a restriction that the "so" is true. In other words, the words "such that" invert the logic. The reader must use the conclusion (that PTP is nonsingular) to figure out what the restriction on P is.

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