Projection Matrix: P=A(ATA)^-1AT & P=BBT

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The discussion centers on the mathematical formulation of projection matrices, specifically P = A(ATA)^-1AT and P = BBT. It is established that for matrix A, its columns must be linearly independent to ensure that ATA is invertible. Additionally, the projection matrix BBT is valid under the condition that A^{-1} equals A^T. The relationship between the dimensions of the column space, row space, and rank is also highlighted as significant in validating the properties of these projection matrices.

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squenshl
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I know that P = A(ATA)-1AT for a projection matrix.
I was just wanting to know how to describe the matrix A as general as possible. For example do the columns and rows of A have to be linearly independent?
Also I know that P = BBT is the projection matrix but how could I describe B as well.
 
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The columns of A have to be linearly independent (so that A^T A is invertable). There's an alternate projection operator for linearly dependent rows. Sounds like B B^T is only a projector if A^{-1} = A^T.
 
Does anyone think the dimension of the column space & the dimension of the rowspace or rank or anything like that has any validation.
 

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