Lecture 3 - 3 Projectors If P Cmm is a square matrix such...

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3 Projectors If P C m × m is a square matrix such that P 2 = P then P is called a projector . A matrix satisfying this property is also known as an idempotent matrix. Remark It should be emphasized that P need not be an orthogonal projection matrix. Moreover, P is usually not an orthogonal matrix. Example Consider the matrix P = c 2 cs cs s 2 , where c = cos θ and s = sin θ . This matrix projects perpendicularly onto the line with inclination angle θ in R 2 . We can check that P is indeed a projector: P 2 = c 2 cs cs s 2 c 2 cs cs s 2 = c 4 + c 2 s 2 c 3 s + cs 3 c 3 s + cs 3 c 2 s 2 + s 4 = c 2 ( c 2 + s 2 ) cs ( c 2 + s 2 ) cs ( c 2 + s 2 ) s 2 ( c 2 + s 2 ) = P. Note that P is not an orthogonal matrix, i.e., P * P = P 2 = P = I . In fact, rank( P ) = 1 since points on the line are projected onto themselves. Example The matrix P = 1 1 0 0 is clearly a projector. Since the range of P is given by all points on the x -axis, and any point ( x, y ) is projected to ( x + y, 0), this is clearly not an orthogonal projection.
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