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Lecture 14 Simplex, Hyper-Cube, Convex Hull and their VolumesPowerPoint Presentation

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 and Subspaces in m-D

- 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)

Simplex

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

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

Determinant of Square Matrix

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

Determinant in 2D

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

Determinant of Square Matrix

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

Properties of Determinant

- 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]

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 a= 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 - tai …] =
det […, ai ,…, aj …] + det […, ai ,…, - tai …]

- det […, ai ,…, aj - tai …] =
- 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|
- (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.

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