# 急! 線性代數的題目

(a) Suppose we have an n × n real-valued matrix Aij which is

known to satisfy

ΣAki*Akj = δij*(aj)^2 (no sum on j)

k＝1

for some positive number aj. Here, δij is the Kronecker delta function

satisfying

..........1 if i ＝ j

δij = {

..........0 if i ≠ j

........︿

│det(Ａ)│ = a1 * a2 * … * an

(Note: It is almost a one-line proof if you use certain properties of

determinants we showed in the class.)

(b) Please give a geometrical interpretation of the result of (a).

"..."是用來對齊算式的

Rating

Qa: A' is the transpose of A

Σ[k=1~n] A(ik)*A(jk) = δ(ij)*[a(j)]^2

ie. A*A'= diag(a(1)^2, a(2)^2, ..., a(n)^2)

=>det (AA')=[det(A)]^2= [a(1)*a(2)*...*a(n) ]^2

=> | det(A) | = a(1)*a(2)*...*a(n)

Qb:

A*A' = diag(a(1)^2, ..., a(n)^2)

means that different row vectors of A are orthogonal, and the module of the jth row equals a(j).

det(A) is the n-dim volume formeb by the n row vectors of A,

so that, V=a(1)*a(2)*a(3)...*a(n)

For example, n=3, three row vectors of A are orthogonal, and with length a(1), a(2), a(3).

These three row vectors form a rectangle box.

The volume of the box equals length*width*height=a(1)a(2)a(3).

thus, different row vectors of A are orthogonal.