Kenneth Kuttler

20.3. p DIMENSIONAL PARALLELEPIPEDS 445

Schmidt procedure there is (w1, · · · ,wp) an orthonormal basis for span(u1, · · · ,up) suchthat span(w1, · · · ,wk) = span(u1, · · · ,uk) for each k ≤ p. We can pick wp =w the givenunit vector perpendicular to each ui. First note that since {wk}p

k=1 is an orthonormal basisfor span(u1, · · · ,up) ,

u j =p

∑k=1

(u j ·wk)wk, u j ·ui =p

∑k=1

(u j ·wk)(ui ·wk)

Therefore, the i jth entry of the p× p matrix UTU is just

(UTU

)i j =

∑r=1

(ui ·wr)(wr ·u j)

which is the product of a p× p matrix M whose r jth entry is wr ·u j with its transpose. Thevector wp is a unit vector perpendicular to each u j for j ≤ p−1 so wp ·u j = 0 if j < p.

Now consider the vector

N ≡ det

w1 · · · wp−1 wp

u1 ·w1 · · · u1 ·wp−1=0

u1 ·wp...

......

up−1 ·w1 · · · up−1 ·wp−1=0

up−1 ·wp

which results from formally expanding along the top row. Note you would get the samething expanding along the last column because as just noted, the last column on the rightis 0 except for the top entry, so every cofactor A1k for the 1kth position is ± a determinantwhich has a column of zeros. Thus N is a multiple of wp. Hence, for j < p,N ·u j = 0.From what was just discussed and induction, v(P(u1, · · · ,up−1)) =±A1p =N ·wp. AlsoN ·up equals

det

up ·w1 · · · up ·wp−1 up ·wp

u1 ·w1 · · · u1 ·wp−1=0

u1 ·wp...

......

up−1 ·w1 · · · up−1 ·wp−1=0

up−1 ·wp

=±det(M)

Thus from induction and expanding along the last column,∣∣up ·wp∣∣v(P(u1, · · · ,up−1)) =

∣∣N ·up∣∣= det

(MT M

)1/2

= det(UTU

)1/2= det(G)1/2 .

Now wp =w the unit vector perpendicular to each u j for j ≤ p− 1. Thus if 20.13, thenthe claimed determinant identity holds.

The theorem shows that the only reasonable definition of p dimensional volume of aparallelepiped is the one given in the above definition. Recall that these vectors are in RM .What is the role of RM? It is just to provide an inner product. That is its only function. Ifp = M, then det

(UTU

)= det

(UT)

det(U) = det(U)2 and so det(G)1/2 = |det(U)|.

20.3. p DIMENSIONAL PARALLELEPIPEDS 445Schmidt procedure there is (w),--- ,w,) an orthonormal basis for span (w1,--- wp) suchthat span (wy),--- ,w,) = span (w1,---, ux) for each k < p. We can pick w, = w the givenunit vector perpendicular to each u;. First note that since {w,; ee , 1s an orthonormal basisfor span (w1,--: , Up),PPuj= V (uj: wy) we, uj uj = Yo (uj: we) (uj we)k=1 k=1Therefore, the ij‘” entry of the p x p matrix U'U is justr P(U'U),,= L (u;-w,) (w,-w;)r=which is the product of a p x p matrix M whose rj"” entry is w,-u j with its transpose. Thevector Wp is a unit vector perpendicular to each u; for j < p—1so wp-uj; =Oif j < p.Now consider the vectorWi tte W p-1 Wp=0Ui *W eee Ul *Wp-l Ul * WpN = det=0Up—-1°W1 +++ Up-1*Wp-1 Up-1°*Wpwhich results from formally expanding along the top row. Note you would get the samething expanding along the last column because as just noted, the last column on the rightis 0 except for the top entry, so every cofactor Aj, for the 1k’” position is + a determinantwhich has a column of zeros. Thus N is a multiple of w,. Hence, for j < p,N-u; =0.From what was just discussed and induction, v(P(w1,---,Up—1)) =+A1p = N- wp. AlsoN - up equalsUp* Wi eee Up*Wp-1 Up* Wp=0Ul, :* WI eee U1 *Wp-1 U1 *Wpdet . . . = +det(M)=0Up—-1°Wi +++ Up-1*Wp-1 Up-1*WpThus from induction and expanding along the last column,|p: wp|v(P(a1,--- 5Up-1)) = |.N - up| = det (M?m)"= det(UTU)"* =det(G)!”.Now w, = w the unit vector perpendicular to each wu; for j < p—1. Thus if 20.13, thenthe claimed determinant identity holds. JjThe theorem shows that the only reasonable definition of p dimensional volume of aparallelepiped is the one given in the above definition. Recall that these vectors are in R”.What is the role of R™? It is just to provide an inner product. That is its only function. Ifp =M, then det (U"U) = det (U") det(U) = det (U)* and so det (G)'/? = |det (U)].