# Absolute values and matrices #

This file proves some bounds on matrices involving absolute values.

## Main results #

• Matrix.det_le: if the entries of an n × n matrix are bounded by x, then the determinant is bounded by n! x^n
• Matrix.det_sum_le: if we have s n × n matrices and the entries of each matrix are bounded by x, then the determinant of their sum is bounded by n! (s * x)^n
• Matrix.det_sum_smul_le: if we have s n × n matrices each multiplied by a constant bounded by y, and the entries of each matrix are bounded by x, then the determinant of the linear combination is bounded by n! (s * y * x)^n
theorem Matrix.det_le {R : Type u_1} {S : Type u_2} [] [] {n : Type u_3} [] [] {A : Matrix n n R} {abv : } {x : S} (hx : ∀ (i j : n), abv (A i j) x) :
abv A.det ().factorial
theorem Matrix.det_sum_le {R : Type u_1} {S : Type u_2} [] [] {n : Type u_3} [] [] {ι : Type u_4} (s : ) {A : ιMatrix n n R} {abv : } {x : S} (hx : ∀ (k : ι) (i j : n), abv (A k i j) x) :
abv (s.sum fun (k : ι) => A k).det ().factorial (s.card x) ^
theorem Matrix.det_sum_smul_le {R : Type u_1} {S : Type u_2} [] [] {n : Type u_3} [] [] {ι : Type u_4} (s : ) {c : ιR} {A : ιMatrix n n R} {abv : } {x : S} (hx : ∀ (k : ι) (i j : n), abv (A k i j) x) {y : S} (hy : ∀ (k : ι), abv (c k) y) :
abv (s.sum fun (k : ι) => c k A k).det ().factorial (s.card y * x) ^