Lecture 14 Simplex, 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

Convex Combination in m-D p1 y p2 p3

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 (1,1,1) (0,1) (0,0,1) (1,0,0) (1,0) n-cube

Pseudo-Hypercube or Pseudo-Box n-Pseudo-Hypercube For any n affinely independent vectors

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

Volume of Pseudo-Hypercube n-Pseudo-Hypercube For any n affinely independent vectors

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

Signed Area and Volume volume( cube(p1,p2) ) = - volume( cube(p2,p1) ) (0,0) p1 volume( cube(p1,p2) ) = - volume( cube(p2,p1) )

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

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

Determinant in 2D Why? p2 =[b,d]T (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 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| (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.