deeplearning.ai 总结 - Gram matrix(格拉姆矩阵)

deeplearning.ai 总结 - Gram matrix(格拉姆矩阵)

flyfish

表示方法
G=VTV G = V T V

解释
A=(abc),B=⎛⎝⎜xyz⎞⎠⎟, A = ( a b c ) , B = ( x y z ) ,

AB=(abc)⎛⎝⎜xyz⎞⎠⎟=ax+by+cz, A B = ( a b c ) ( x y z ) = a x + b y + c z ,
BA=⎛⎝⎜xyz⎞⎠⎟(abc)=⎛⎝⎜xayazaxbybzbxcyczc⎞⎠⎟. B A = ( x y z ) ( a b c ) = ( x a x b x c y a y b y c z a z b z c ) .

行列式可表示为
G(x1,…,xn)=∣∣∣∣∣∣∣⟨x1,x1⟩⟨x1,x2⟩…⟨x1,xn⟩⟨x2,x1⟩⟨x2,x2⟩…⟨x2,xn⟩⋮⋮⋱⋮⟨xn,x1⟩⟨xn,x2⟩…⟨xn,xn⟩∣∣∣∣∣∣∣ G ( x 1 , … , x n ) = | ⟨ x 1 , x 1 ⟩ ⟨ x 1 , x 2 ⟩ … ⟨ x 1 , x n ⟩ ⟨ x 2 , x 1 ⟩ ⟨ x 2 , x 2 ⟩ … ⟨ x 2 , x n ⟩ ⋮ ⋮ ⋱ ⋮ ⟨ x n , x 1 ⟩ ⟨ x n , x 2 ⟩ … ⟨ x n , x n ⟩ |

另一种表示
G(x1,…,xn)=∥x1∧⋯∧xn∥2. G ( x 1 , … , x n ) = ‖ x 1 ∧ ⋯ ∧ x n ‖ 2 .