This is from my linear regression class i attached the...

This is from my linear regression class. i attached the definition in the image. Thanks a lot for your help ! 3.7 Suppose that A is a p p symmetric matrix that we write in partitioned form A11 A12 A12 A22 A The matrix Au is pi p1, so A22 is (p - pi) (p - pi). One can show that if A-1 exists, it can be written as Al+A12A22Ai2-A12A22 =AA2 A22 Using this result, show that, if X is an n (p + 1) data matrix with all 1s in the first column, 1+x'(x'x)-1x-x'(x'x)-1 -(X'x)-1x (x'x)- X'x x-xx1p-x (x1-x)...(x2p-x) definition:= : : : (xn1-x)...(xnp-xp) where and x are defined in Section 3.4.3

Transcript

00:01 Hello students, let's discuss the question.

00:04 Here, consider a matrix with linearly independent columns.

00:08 Now we have to prove that p square is equal to p by multiplying p by itself and simplified.

00:16 Here, p is the projection matrix.

00:27 P is equal to a multiplied by transpose of a multiplied by a, whole inverse multiplied by a, transpose of a multiplied by a, transpose of a, of a.

00:41 Now here p square is equal to p multiplied by p so p is equal to a multiplied by transpose a multiplied by a whole inverse a transpose now this is again multiplied by by a multiplied by transpose a again a inverse multiplied by transpose a now this transpose a and a is nothing but an identity matrix.

01:24 So a multiplied by identity matrix.

01:28 Now this is again and this can be written as a multiplied by a transpose multiplied by a inverse.

01:42 Again this is equal to identity and transpose a.

01:50 Now further, a multiplied by this is identity matrix transpose a.

02:06 This is equal to a multiplied by a transpose a transpose a, whole inverse a transpose.

02:16 Now look at this, this is nothing but p.

02:20 So p square is equal to p.

02:23 Here we have proved the first part.

02:26 Now in the next part we have to prove that p is a symmetric by computing transpose of p.

02:35 So transpose of p is equal to a multiplied by a transpose a inverse multiplied by a transpose whole transpose.

02:48 This is equal to now transpose a again it's transpose multiplied by a multiplied by a.

03:00 A transpose a, whole inverse, whole transpose multiplied by a transpose.

03:10 Now further computing we'll get...

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