行列式 与 n维平行多面体体积 公式的证明 determinant volume proof

Q: why determinant is volume of parallelepiped in any dimensions？

A: http://math.stackexchange.com/a/547522/227287 

Suppose v1,v2,...vn are linear independent.

In 2d,

                                        

The signed area (2-volume) of (v1,v2) is vol(v1,v2).

v1=(x,0) v2'=(0,y)

v2' = v2 - C*v1.

vol(v1,v2) = vol(v1,v2') = x*y = det(v1,v2')= det(v1,v2 - C*v1)= det(v1,v2)

In 3d,

The signed volume (3-volume) of (v1,v2,v3) is vol(v1,v2,v3).

v1=(x,0,0) v2'=(0,y,0) v3'=(0,0,z)

v2' = v2 - C21*v1.

v3' = v3 - C31*v1 - C32*v2.

vol(v1,v2,v3) = vol(v1,v2',v3') = x*y*z = det(v1,v2',v3')

= det(v1, v2-C21*v1, v3-C31*v1-C32*v2)= det(v1, v2, v3-C31*v1-C32*v2)

= det(v1,v2,v3)

...

In n-d:

(signed) n-volume of (v1,v2,...,vn) is vol(v1,v2,...,vn).

v1=(x1,0,...,0) v2'=(0,x2,...,0) ... vn'=(0,...,0,xn)

v2' = v2 - C21*v1.

v3' = v3 - C31*v1 - C32*v2.

...

vn' = vn - Cn1*v1 - Cn2*v2 -...- Cnn-1*vn-1

vol(v1,v2,...,vn) = vol(v1,v2',...,vn') = x1*x2*...*xn = det(v1,v2',...,vn')

= det(v1, v2 - C21*v1, ..., vn - Cn1*v1 - Cn2*v2 -...- Cnn-1*vn-1)

= det(v1,v2,..., vn-1, vn - Cn1*v1 - Cn2*v2 -...- Cnn-1*vn-1)

= det(v1,v2,..., vn)

if v1,v2 are linear dependent, det(v1,v2)=0 and vol(v1,v2)(which is signed area)=0.

if v1,v2,v3 are linear dependent, det(v1,v2,v3)=0 and vol(v1,v2,v3)(which is signed volume)=0.

if v1,v2, ..., vn are linear dependent, det(v1,v2,...,vn)=0 and vol(v1,v2,...,vn)=0.

So we always have vol(v1,v2,...,vn) = det(v1,v2,..., vn).

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Determinant : Scaling factor of n-volume

Matrix A = (a1,a2,...,an), ai is n-d vector.

E =(e1,e2,...,en) is the identity matrix.

AE = A.

A(e1,e2,...,en) = (a1,a2,...,an).

A is a linear transform, (e1,e2,...,en) --A-> (a1,a2,...,an)

vol(e1,e2,...,en) = 1.

vol(a1,a2,...,an) = det(A).

So, the scaling factor of Matrix A is det(A).

if n=1, it's a scaling factor of signed length(1-volume).

if n=2, it's a scaling factor of signed area(2-volume).

if n=3, it's a scaling factor of signed volume(3-volume).

...

For any invertible Matrix M (a n-parallelepiped) and its transformation AM:

scaling factor of A = vol(AM)/vol(M) = det(AM)/det(M) = det(A)det(M) /det(M) = det(A).

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