Orthogonal
more to say
- you see | number |, you see || vector ||
notation of
another way to denote inner prodoct, i.e. dot product
def. of euclidean length
distance from origin to arror end. Sum of each component’s square and take square root
def. of distance
distance of arbitary two vector
scalar product of vector length
simply times a scalar
def. vector in
inner product is zero
勾股定理
same in geometric
def. of projection with dir. vector
orthogonal subspace in
any vector from V denote as v, similar W. Have property that for all v in V and w in W.
a plane and its perp line
Orthogonal complement and its properties
Fundamental subspace theorem
def. of orthogonal
Lemma 19.5 (Cauchy-Schwartz Inequality)
least quare solution
normal equations
最小二乘
inner product in more general way
denote as , x and y are vector in vector space V, to be more particular, called it “inner product space”.
general inner product must fufill 3 properties as follows:
Frobenius inner product (inner product for matrix)
definition
For matrices and of the same dimensions, the Frobenius inner product is defined as:
Where is the trace of a matrix.
这个看不懂,但是可以写成矩阵相同位置一一相乘的形式
length in inner product space
a vector v in inner product space, length of v is denote as
orthorgonal in inner product space
seems that we define a new way for inner produc
in previews learning, we define inner product as , which is traditional inner product taught in high school
but now inner product can be define in another way
so correspondingly we have a more general definition of what inner product is. And what orthogonal is.(since it is defined as <x,y> = 0)
Theo. 勾股定理 in inner VS
垂直有0,产生的等式
Theo. 21.5 Cauchy inequality
normed vector space
A vector space V is said to be a normed linear space if
each vector v ∈ V is associated with a real
number ||v||∈ R, called the norm of v, satisfying:
Theo 21.7 norm on inner product space
Orthogonal set
for in inner product space V, any , that , then the set is orthogonal set
Orthonomal set
in Orthogonal set, take all the vactor with norm == 1, then it is orthonormal set.
Theo. 21.10 Orthogonal set is linear indep.
def. orthonomal basis
- B is the basis if B is an orthonomal set of V
- Span{B} = V
coor. w.r.t orthonomal basis
any V can be decompose
def. 21.25 orthogonal matrix
if column vector in Q is orthonomal set in , then Q is orthogonal matrix
*好严苛的判定
equivalent condition for orthogonal matrix
prop. 5 properties that orthogonal matrix have
Gram Schmidt Process 格拉姆-施密特
对 中任意非零子空间,快速求解,正交基/标准正交基 的方法