Lecture 14 simplex hyper cube convex hull and their volumes
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Lecture 14 Simplex, Hyper-Cube, Convex Hull and their Volumes. Shang-Hua Teng. Linear Combination and Subspaces in m-D. Linear combination of v 1 (line) { c v 1 : c is a real number} Linear combination of v 1 and v 2 (plane) { c 1 v 1 + c 2 v 2 : c 1 ,c 2 are real numbers}

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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

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

Shang-Hua Teng


Linear combination and subspaces in m d

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)


Affine combination in m d

Affine Combination in m-D


Convex combination in m d

Convex Combination in m-D

p1

y

p2

p3


Simplex

Simplex

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


Parallelogram

Parallelogram


Parallelogram1

Parallelogram


Hypercube

Hypercube

(1,1,1)

(0,1)

(0,0,1)

(1,0,0)

(1,0)

n-cube


Pseudo hypercube or pseudo box

Pseudo-Hypercube or Pseudo-Box

n-Pseudo-Hypercube

For any n affinely independent vectors


Convex set

Convex Set


Non convex set

Non Convex Set


Convex set1

Convex Set

A set is convex if the line-segment between

any two points in the set is also in the set


Non convex set1

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

Convex Hull

Smallest convex set that contains all points


Convex hull1

Convex Hull


Volume of pseudo hypercube

Volume of Pseudo-Hypercube

n-Pseudo-Hypercube

For any n affinely independent vectors


Properties of volume of n d pseudo hypercube in n d

Properties of Volume of n-D Pseudo-Hypercube in n-D


Signed area and volume

Signed Area and Volume

p2

(0,0)

p1

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


Rule of signed volume n d pseudo hypercube in n d

Rule of Signed Volume n-D Pseudo-Hypercube in n-D


Determinant of square matrix

Determinant of Square Matrix

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


Determinant in 2d

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 matrix1

Determinant of Square Matrix

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


Properties of determinant

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

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 it1

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 …]

  • One can compute determinant by elimination

    • PA = LU then det A = det U


Properties of determinant and algorithm for computing it2

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

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

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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