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Lecture 14Simplex, Hyper-Cube, Convex Hull
and their Volumes
Shang-Hua Teng
Linear Combination and Subspaces in m-D
• Linear combination of v1 (line)
{c v1 : c is a real number}
• Linear combination of v1 and v2 (plane)
{c1 v1 + c2 v2 : c1 ,c2 are real numbers}
• Linear combination of n vectors v1 , v2 ,…, vn
(n Space)
{c1v1 +c2v2+…+ cnvn : c1,c2 ,…,cn are real numbers}
Span(v1 , v2 ,…, vn)
Affine Combination in m-D
1
:
1
,,,affine
21
2211
1
12211
21
n
nn
n
n
j j
n
ppp
ppp
ppp
Convex Combination in m-D
ni
ppp
ppp
n
nn
n
1 ,0
1
:
,,,Convex
i
21
2211
21
y
p1
p2p3
Simplex
n dimensional simplex in m dimensions (n < m) is the set of all convex combinations of n + 1 affinely independent vectors
Parallelogram
Parallelogram
Hypercube
n-cube
Ti
nn
e
nic
ececec c
010
1 ,1,0
:0convex
i
22110
(1,0)
(0,1)
(1,0,0)
(0,0,1)
(1,1,1)
Pseudo-Hypercube or Pseudo-Box
n-Pseudo-HypercubeFor any n affinely independent vectors
nic
pcpcpccppp nn
n1 ,1,0
:0 convex ,,, cube
i
2211021
nppp ,,, 21
Convex Set
Non Convex Set
Convex Set
A set is convex if the line-segment between any two points in the set is also in the set
Non Convex Set
A set is not convex if there exists a pair of points whose line segment is not completely in the set
Convex Hull
Smallest convex set that contains all points
Convex Hull
nppp ,,,Convex 21
Volume of Pseudo-Hypercube
n-Pseudo-HypercubeFor any n affinely independent vectors nppp ,,, 21
nic
pcpcpccppp nn
n1 ,1,0
:0 convex ,,, cube
i
2211021
Properties of Volume of n-D Pseudo-Hypercube in n-D
1 ,,, cubevolume 21 neee
,,, cube volume ,,, cubevolume 2121 nn pppppp
,,, cube volume ,,, cubevolume 2121 nn
n pppppp
Signed Area and Volume
(0,0)
p1
p2
volume( cube(p1,p2) ) = - volume( cube(p2,p1) )
Rule of Signed Volume n-D Pseudo-Hypercube in n-D
,,,, cube volume-
,,,, cubevolume
1
1
nij
nji
pppp
pppp
Determinant of Square Matrix
,,, cubevolume
,,, ,,,det
21
2121
n
nn
ppp
pppppp
How to compute determinant or the volume of pseudo-cube?
Determinant in 2D
ad-bcdc
ba det
(0,0)
p1 =[a,c]T
p2 =[b,d]T
Why?
ac
bd
dc
baac
bd
ad-bcdc
ba 11
1
Invertible if and only if the determinant is not zeroif and only if the two columns are not linearly dependent
Determinant of Square Matrix
,,, cubevolume
,,, ,,,det
21
2121
n
nn
ppp
pppppp
How to compute determinant or the volume of pseudo-cube?
Properties of Determinant1. det I = 12. The determinant changes sign when sign when two
rows are changed (sign reversal)1. Determinant of permutation matrices are 1 or -1
3. The determinant is a linear function of each row separately
1. det [a1 , …,tai ,…, an] = t det [a1 , …,ai ,…, an] 2. det [a1 , …, ai + bi ,…, an] = det [a1 , …,ai ,…, an] + det [a1 , …, bi ,…, an] • [Show the 2D geometric argument on the board]
Properties of Determinant and Algorithm for Computing it
• [4] If two rows of A are equal, then det A = 0– Proof: det […, ai ,…, aj …] = - det […, aj ,…, ai …]
– If ai = aj then
– det […, ai ,…, aj …] = -det […, ai ,…, aj …]
Properties of Determinant and Algorithm for Computing it
• [5] Subtracting a multiple of one row from another row leaves det A unchanged
– det […, ai ,…, aj - t ai …] =
det […, ai ,…, aj …] + det […, ai ,…, - t ai …]
• One can compute determinant by elimination– PA = LU then det A = det U
Properties of Determinant and Algorithm for Computing it
• [6] A matrix with a row of zeros has det A = 0
• [7] If A is triangular, then– det [A] = a11 a22 … ann
• The determinant can be computed in O(n3)
time
Determinant and Inverse
• [8] If A is singular then det A = 0. If A is invertible, then det A is not 0
Determinant and Matrix Product
• [9] det AB = det A det B (|AB| = |A| |B|)– Proof: consider D(A) = |AB| / |B|1. (Determinant of I) A = I, then D(A) = 1.2. (Sign Reversal): When two rows of A are exchanged,
so are the same two rows of AB. Therefore |AB| only changes sign, so is D(A)
3. (Linearity) when row 1 of A is multiplied by t, so is row 1 of AB. This multiplies |AB| by t and multiplies the ratio by t – as desired.
