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Why don we just use the formula P=AAT to find the matrix P of the orthogonal projection onto a subspace V?

April 26, 2017Formula matrix orthogonal p projection subspace v

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Why don we just use the formula P=AAT to find the matrix P of the orthogonal projection onto a subspace V?

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the formula P=A AT gives the orthogonal projection if the column vectors of A are orthonormal vectors only. In general one has to use a formula which was derived in least square solutions: P= A (AT A)-1 AT which simplifies in the case of orthonormal columns to A AT because the middle part is then an identity matrix.