The Veronese map of degree 2 is a mapping from R n + 1 {\displaystyle \mathbb {R} ^{n+1}} to the space of symmetric matrices ( n + 1 ) × ( n + 1 ) {\displaystyle (n+1){\times }(n+1)} defined by the formula:
V : ( x 0 , … , x n ) → ( x 0 ⋅ x 0 x 0 ⋅ x 1 … x 0 ⋅ x n x 1 ⋅ x 0 x 1 ⋅ x 1 … x 1 ⋅ x n ⋮ ⋮ ⋱ ⋮ x n ⋅ x 0 x n ⋅ x 1 … x n ⋅ x n ) . {\displaystyle V\colon (x_{0},\dots ,x_{n})\to {\begin{pmatrix}x_{0}\cdot x_{0}&x_{0}\cdot x_{1}&\dots &x_{0}\cdot x_{n}\\x_{1}\cdot x_{0}&x_{1}\cdot x_{1}&\dots &x_{1}\cdot x_{n}\\\vdots &\vdots &\ddots &\vdots \\x_{n}\cdot x_{0}&x_{n}\cdot x_{1}&\dots &x_{n}\cdot x_{n}\end{pmatrix}}.}
Note that V ( x ) = V ( − x ) {\displaystyle V(x)=V(-x)} for any x ∈ R n + 1 {\displaystyle x\in \mathbb {R} ^{n+1}} . In particular, the restriction of V {\displaystyle V} to the unit sphere S n {\displaystyle \mathbb {S} ^{n}} factors through the projective space R P n {\displaystyle \mathbb {R} \mathrm {P} ^{n}} , which defines the Veronese embedding of R P n {\displaystyle \mathbb {R} \mathrm {P} ^{n}} . The image of the Veronese embedding is called the Veronese submanifold, and for n = 2 {\displaystyle n=2} it is known as the Veronese surface.
Properties The matrices in the image of the Veronese embedding correspond to projections onto one-dimensional subspaces in R n + 1 {\displaystyle \mathbb {R} ^{n+1}} . They can be described by the equations:
A T = A , t r A = 1 , A 2 = A . {\displaystyle A^{T}=A,\quad \mathrm {tr} \,A=1,\quad A^{2}=A.}
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