Question1: Consider a QR faction M=QR, show that R= Transpose(Q)M You need to show that (1)M...

Question1: Consider a QR faction M=QR, show that R= Transpose(Q)M

You need to show that (1)M = QR where R := Transpose(Q)M and (2) that R is upper triangular.

To show (1) use the fact that QTranspose(Q) is the matrix for orthogonal projection onto the image of M. What happens to a column of M (which is a vector in the image of M) when you project it onto the image of M?

To show (2), think about the entries of R := Transpose(Q)M as dot products between the columns v_1,...,v_n of M and the rows u_1,...,u_n of Q^T. Entries of Transpose(Q)M vanish when these vectors are orthogonal. The vectors u_1,..., u_n are the othonomal basis for the image of M obtained from v_1,...,v_n via the Gram-Schmidt process. Why is it the case that u_i.v_j =0 if i>j?

Expert Solution

Colby Messinger answered 2 months ago

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