18.06.23: Determinants & Permutations Lecturer: Barwick Friday, 8 April 2016 18.06.23: Determinants & Permutations So here’s an 푛 × 푛 matrix 퐴 = ( ~푣1 ⋯ ~푣푛 ), and we have this number det(퐴) = det (~푣1,…,~푣푛) ∈ R that measures the signed 푛-dimensional volume of the parallelopiped spanned by ~푣1,…,~푣푛. Let’s list the things we know about det(퐴). 18.06.23: Determinants & Permutations (1) Normalization. The identity matrix has determinant 1: det(퐼) = det (푒1 ̂ , ⋯ ,푒푛 ̂ ) = 1. (2) Multilinearity. For any real numbers 푟, 푠 ∈ R, det (~푣1, ⋯ , 푟~푥푖 + 푠~푦푖, ⋯ ,~푣푛) = 푟 det (~푣1, ⋯ ,~푥푖, ⋯ ,~푣푛) +푠 det (~푣1, ⋯ ,~푦푖, ⋯ ,~푣푛). 18.06.23: Determinants & Permutations (3) Alternation. The determinant det (~푣1, ⋯ , ⋯ ,~푣푛) = 0 if any two of the ~푣푖s are equal. These are the core, defining properties of det. 18.06.23: Determinants & Permutations Here are more properties, which we deduced from the three core properties above: (4) Multiplying a row or column by a number 푟 ∈ R multiplies the determi- nant by that 푟. (5) Swapping two rows or columns in 퐴 multiplies the determinant by a −1. (6) Adding a multiple of a row or column onto another row or column doesn’t change the determinant. 18.06.23: Determinants & Permutations And here are some general computational facts we extract from these proper- ties. (8) det(퐴) ≠ 0 if and only if 퐴 is invertible. (9) det(푀푁) = det(푀) det(푁). (10) det(퐴⊺) = det(퐴). (Why?) 푛 (11) det diag(휆1,…, 휆푛) = ∏푖=1 휆푖. (12) More generally, if 퐴 is triangular, then det(퐴) is the product of the entries along the diagonal. (Why?) 18.06.23: Determinants & Permutations Let’s compute the determinant of the following matrices: 1 1 1 2 1 2 4 1 1 2 2 ( 1 3 9 ),( ) 1 2 2 3 1 4 16 2 2 3 4 1 1 2 5 1 2 5 14 1 2 5 14 2 5 5 14 ( ),( ) 2 5 14 42 5 14 42 132 5 14 42 132 14 42 132 429 18.06.23: Determinants & Permutations I was half-joking about the last two; this is actually a neat little piece of math- ematics: there’s only one sequence of integers 푐0, 푐1, 푐2,… such that for any 푛 ≥ 1, det(푐푖+푗−2) = det(푐푖+푗−1) = 1. These are called the Catalan numbers, and your mathematical life isn’t com- plete until you’ve read about them! (I was going to put a problem about these on the homework, but I thought that might be too much.) 18.06.23: Determinants & Permutations Suppose 휎∶ {1, … , 푛} {1, … , 푛} a permutation of {1, … , 푛} – i.e., a bijection from that set to itself. One can express a permutation very compactly, by writing down the matrix 푃휎 = ( 푒휎̂ (1) ⋯푒휎 ̂ (푛) ) called the permutation matrix corresponding to 휎. This is great, because matrix multiplication corresponds to composition of permutations: 푃휎∘휏 = 푃휎푃휏 and 푃푖푑 = 퐼. 18.06.23: Determinants & Permutations Also, these matrices are orthogonal; in fact, ⊺ −1 푃휎 = 푃휎 = 푃휎−1 . 18.06.23: Determinants & Permutations Here’s a permutation matrix for 푛 = 5: 푒2̂ 0 1 0 0 0 푒4̂ 0 0 0 1 0 ( ) ( ) 푒1̂ = 1 0 0 0 0 . 푒3̂ 0 0 1 0 0 ( 푒5̂ ) ( 0 0 0 0 1 ) What’s its determinant? 18.06.23: Determinants & Permutations This is a general pattern: the determinant of a permutation matrix 푃휎 is called the sign of the permutation 휎: sgn(휎) ≔ det(푃휎) In effect, it’s (−1)number of swaps in 휎. What’s weird about this is that you can imagine performing more or fewer swaps to get sigma. The magic of determinants is telling you that the parity of the number of swaps stays the same! Note that sgn(휎) = sgn(휎−1). (Why?) 18.06.23: Determinants & Permutations It turns out that permutations give you a formula for the determinant of any matrix 퐴 = ( ~푣1 ⋯ ~푣푛 ). Let’s see why. First, the 푗-th column ~푣푗 can be written as 푛 ~푣푗 = ∑ 푎푘,푗푒푘̂ . 푘=1 18.06.23: Determinants & Permutations The multilinearity of det can then be deployed: 푛 푛 det(퐴) = det ( ∑ 푎푘(1),푗푒푘̂ (1),…, ∑ 푎푘(푛),푗푒푘̂ (푛)) 푘(1)=1 푘(푛)=1 푛 푛 푛 = ∑ ⋯ ∑ (∏ 푎푘(푖),푖) det(푒푘 ̂ (1), … ,푒푘 ̂ (푛)) 푘(1)=1 푘(푛)=1 푖=1 Now all those sums can be combined into one sum. You’re summing over the set 퐸푛 of all maps 푘∶ {1, … , 푛} {1, … , 푛}: 푛 det(퐴) = ∑ (∏ 푎푘(푖),푖) det(푒푘 ̂ (1), … ,푒푘 ̂ (푛)). 푘∈퐸푛 푖=1 18.06.23: Determinants & Permutations 푛 det(퐴) = ∑ (∏ 푎푘(푖),푖) det(푒푘 ̂ (1), … ,푒푘 ̂ (푛)). 푘∈퐸푛 푖=1 Now we use the alternatingness: if any two columns are equal, then the de- terminant is zero. So any summand in which 푘∶ {1, … , 푛} {1, … , 푛} is not injective doesn’t appear: 푛 det(퐴) = ∑ (∏ 푎휎(푖),푖) det(푒휎 ̂ (1), … ,푒휎 ̂ (푛)), 휎∈훴푛 푖=1 where 훴푛 is the set of permutations of {1, … , 푛}. 18.06.23: Determinants & Permutations Now we have: 푛 det(퐴) = ∑ (∏ 푎휎(푖),푖) det(푒휎 ̂ (1), … ,푒휎 ̂ (푛)) 휎∈훴푛 푖=1 푛 = ∑ (∏ 푎휎(푖),푖) det(푃휎) 휎∈훴푛 푖=1 푛 = ∑ sgn(휎)(∏ 푎휎(푖),푖). 휎∈훴푛 푖=1 18.06.23: Determinants & Permutations 푛 det(퐴) = ∑ sgn(휎)(∏ 푎휎(푖),푖) 휎∈훴푛 푖=1 There it is – the Leibniz formula for the determinant. Do you care? Well, if you’re trying to program a computer to compute deter- minants, no. Evaluating this formula involves 훺(푛! 푛) operations. Gaussian elimination uses 푂(푛3) operations. We have a winner. On the other hand, the fact that there is a formula is vaguely reassuring. But there’s another advantage … 18.06.23: Determinants & Permutations 2 Think of the determinant as a function from R푛 R; this formula expresses that function as a polynomial in 푛2 variables. That means that it’s continuous and infinitely differentiable. So this leads us to the following result: Proposition. Suppose 퐴 = (푎푖,푗) an invertible 푛 × 푛 matrix. Then there exists ′ ′ ′ an 휀 > 0 such that if 퐴 = (푎푖,푗) is an 푛 × 푛 matrix such that |푎푖,푗 − 푎푖,푗| < 휀, then 퐴′ is invertible too. That is, invertible matrices are stable under small perturbations. 18.06.23: Determinants & Permutations 2 Here’s another wacky-sounding consequence. Suppose 퐿 ⊆ R푛 is a line. Then if there exists one point on 퐿 that corresponds to an invertible matrix, then all but finitely many points on 퐿 correspond to invertible matrices. 18.06.23: Determinants & Permutations Question. Suppose 퐴 an 푛 × 푛 matrix. For how many real numbers 푡 ∈ R is 퐴 + 푡퐼 is invertible (none, finitely many, infinitely many, all)?.