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Chapter 5: Q42E (page 234)

Let Abe the matrix of an orthogonal projection. FindA2 in two ways:

a.Geometrically. (Consider what happens when you apply an orthogonal projection twice.)

b.By computation, using the formula given in Theorem 5.3.10

Short Answer

Expert verified

A2=A

Step by step solution

01

Consider for part (a).

Observe that, in the figure below the brown vector is the orthogonal projection of the vector v鈯�onto the space W. so, if we project onto a subspace W of Rn, then for any xI鈭�Rn,AxI鈭�W. Now, again taking the orthogonal projection won鈥檛 do anything as the vector Ax鈯�is already there on the subspace W and therefore, A(AxI)=AxI=A2xI

Since it is true for any xI鈭�Rnthus,

A2=A

Consider the diagram below.

02

Consider for part (b).

Observe that the matrix of the orthogonal projection is given by .

As the ijth entry of the matrix is given by

uirujr=0i鈮�j1i鈮�j

So, consider, localid="1659500792037" A=QQT鈬�A2

A=QQT鈬�A2=QQTQQT=(QQT)(QQT)=QQT=A

Thus, it givesA2=A

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If Ais ann脳mmatrix, is the formulaim(A)=im(AAT)necessarily true? Explain.

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