The inner product of two column vectors of size n×1, u, v∈Rn is defined as follows.
u⋅v:=uTv=u1v1+u2v2+⋯+unvn
In the case where u, v∈Cn, it is as follows.
u⋅v:=u∗v=u1∗v1+u2∗v2+⋯+un∗vn
Here, u is the conjugate transpose of u. Two vectors u, v are said to be orthogonal to each other if they satisfy the following equation, and it is denoted as u⊥v.
u⋅v=0
The norm or length of the column vector v is defined as follows.
∥v∥:=v⋅v
A vector with norm 1 is called a unit vecter. The distance between two vectors u, v is represented as d(u.v) and defined as follows.
d(u,v):=∥u−v∥=(u−v)⋅(u−v)=(u−v)∗(u−v)
Explanation
In coordinate space, the inner product of two vectors is nothing but its representation as a matrix multiplication and its expansion to complex numbers. Therefore, (EuclideanIP) is called the Euclidean inner product or standard inner product. Thus, the notation ⋅ is used for the inner product, but the general notation for the inner product is as follows.
⟨u,v⟩
By definition, in the case of real matrices, u⋅v=v⋅u holds, and in the case of complex matrices, u⋅v=v⋅u holds.
The core concept of the inner product is ‘multiply the same components and sum them all,’ which generalized for n×n matrix is as follows.
Properties
Let A be a n×n real matrix and u,v a n×1 real matrix. Then, the following equation holds.
Au⋅vu⋅Av=u⋅ATv=ATu⋅v
In the case of complex matrices, the following equation holds.
Au⋅vu⋅Av=u⋅A∗v=A∗u⋅v
Proof
Since the method of proving the four formulas is the same, only the proof of the first formula is introduced.