Lecture 14 Simplex, Hyper-Cube, Convex Hull and their Volumes

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Lecture 14 Simplex, Hyper-Cube, Convex Hull and their Volumes

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Lecture 14 Simplex, Hyper-Cube, Convex Hull and their Volumes

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Lecture 14Simplex, Hyper-Cube, Convex Hull and their Volumes

Shang-Hua Teng

- Linear combination of v1(line)
{c v1: c is a real number}

- Linear combination of v1and v2(plane)
{c1v1 + c2v2: c1 ,c2are real numbers}

- Linear combination of n vectors v1 , v2 ,…, vn
(n Space)

{c1v1 +c2v2+…+ cnvn: c1,c2 ,…,cnare real numbers}

Span(v1 , v2 ,…, vn)

p1

y

p2

p3

n dimensional simplex in m dimensions (n < m) is the set of all convex combinations of n + 1 affinely independent vectors

(1,1,1)

(0,1)

(0,0,1)

(1,0,0)

(1,0)

n-cube

n-Pseudo-Hypercube

For any n affinely independent vectors

A set is convex if the line-segment between

any two points in the set is also in the set

A set is not convex if there exists a pair of points

whose line segment is not completely in the set

Smallest convex set that contains all points

n-Pseudo-Hypercube

For any n affinely independent vectors

p2

(0,0)

p1

volume( cube(p1,p2) ) = - volume( cube(p1,p2) )

How to compute determinant or the volume of pseudo-cube?

p2 =[b,d]T

Why?

(0,0)

p1 =[a,c]T

Invertible if and only if the determinant is not zero

if and only if the two columns are not linearly dependent

How to compute determinant or the volume of pseudo-cube?

- det I = 1
- The determinant changes sign when sign when two rows are changed (sign reversal)
- Determinant of permutation matrices are 1 or -1

- The determinant is a linear function of each row separately
- det [a1 , …,tai ,…, an] = t det [a1 , …,ai ,…, an]
- det [a1 , …, ai+ bi ,…, an] =
det [a1 , …,ai ,…, an] + det [a1 , …, bi ,…, an]

- [Show the 2D geometric argument on the board]

- [4] If two rows of A are equal, then det A = 0
- Proof: det […, ai ,…, aj …] = - det […, aj ,…, ai …]
- If a= aj then
- det […, ai ,…, aj …] = -det […, ai ,…, aj …]

- [5] Subtracting a multiple of one row from another row leaves det A unchanged
- det […, ai ,…, aj - tai …] =
det […, ai ,…, aj …] + det […, ai ,…, - tai …]

- det […, ai ,…, aj - tai …] =
- One can compute determinant by elimination
- PA = LU then det A = det U

- [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

- [8] If A is singular then det A = 0. If A is invertible, then det A is not 0

- [9] det AB = det A det B (|AB| = |A| |B|)
- Proof: consider D(A) = |AB| / |B|
- (Determinant of I) A = I, then D(A) = 1.
- (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)
- (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.