Orthogonality and Projection

悠悠 2022-04-17 03:28 247阅读 0赞

### 文章目录 ###

*   *  Orthogonality of the four subspaces
     *   *  Definition of orthogonal subspace
         *  Definition of orthogonal complements(正交补)
     *  Projections
     *   *  Projection onto a line
         *  Projection onto a subspace
     *  Orthogonal bases and Gram-Schmidt
     *   *  Orthonomal bases and matrix(标准正交基、正交矩阵)
         *  Projection Using Orthonormal Bases:Q replace A
         *  The Gram-Schmidt

## Orthogonality of the four subspaces ##

![在这里插入图片描述][20181112155247602.png]

*  The row space is perpendicular to the nullspace
 *  The column space is perpendicular to the nullspace of  A T A^T AT  
      
    
 *  The column space C(A) , a subspace of  R m R^m Rm, dimention r
 *  The leftnull space N( A T A^T AT) , a subspace of  R m R^m Rm ,dimention m-r
 *  The row space C( A T A^T AT) , a subspace of  R n R^n Rn ,dimention r
 *  The null space N(A) , a subspace of  R n R^n Rn ,dimention n-r

### Definition of orthogonal subspace ###

*  Two subspace  V V V and  W W W of a vector space are orthogonal if every vector  v v v in  V V V is perpendicular to every vector  w w w in  W W W

*  Every vector x in the nullspace is perpendicular to every row of  A A A, because  A x = 0 Ax=0 Ax=0 ,the nullspace  N ( A ) N(A) N(A) and row space  C ( A T ) C(A^T) C(AT) are orthogonal subspace of  R n R^n Rn
 *  Every vector  y y y in the nullspace of  A T A^T AT is perpendicular to every column of  A A A, because  A T x = 0 A^Tx=0 ATx=0 , ,The leftnull space N( A T A^T AT) and the column space C(A) are orthogonal in  R m R^m Rm

### Definition of orthogonal complements(正交补) ###

*  The orthogonal complement of a subspace  V V V contains every vector that is perpendicular to  V V V

*   Nullspace N(A) is the orthogonal complement of the row space C( A T A^T AT) in  R n R^n Rn
 *   LeftNull space N( A T A^T AT) is the orthogonal complement of the column space C(A) in  R m R^m Rm

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## Projections ##

*   The projection matrix P（由A得到投影矩阵） multiplies b（被投影的向量） to give p (b在A上的投影) on A

### Projection onto a line ###

![在这里插入图片描述][2018111217192718.png]

projection matrix : P = a a T a T a P=\\frac\{aa^T\}\{a^Ta\} P=aTaaaT

例子：  
![在这里插入图片描述][20181112193334180.png]

### Projection onto a subspace ###

![在这里插入图片描述][20181112192700526.png]  
![在这里插入图片描述][20181112192206447.png]

( I − P ) b = e (I-P)b=e (I−P)b=e : I − P I-P I−P 也是投影矩阵，将b 投影到 plane perpendicular to A to get e

实例  
![在这里插入图片描述][20181112193639979.png]

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## Orthogonal bases and Gram-Schmidt ##

### Orthonomal bases and matrix(标准正交基、正交矩阵) ###

*  标准正交基  
    ![\- Definition:][- Definition]
 *  (orthogonal matrix)正交矩阵:方阵

由上面标准正交基构成的方阵：正交矩阵  
![在这里插入图片描述][20181112200052273.png]

the inverse is the transpose , In the square case we call Q an orthogonal matrix

### Projection Using Orthonormal Bases:Q replace A ###

*  Suppose the basis are orthonomal. that means A is Q ,  A T A A^TA ATA is  Q T Q = I Q^TQ=I QTQ=I  
    投影向量： p = Q x ^ = Q Q T b p=Q\\hat\{x\}=QQ^Tb p=Qx^=QQTb  
     x ^ = Q T b \\hat\{x\}=Q^Tb x^=QTb  
    projection matrix: P = Q Q T P=QQ^T P=QQT

![在这里插入图片描述][20181112211012187.png]

*   When Q is square m=n , (subspace Q is whole space, 任何一个向量投影到整个整个空间，都是其本身，所以投影矩阵是 I I I)  
     Q T = Q − 1 Q^T=Q^\{-1\} QT=Q−1  
     x ^ = Q − 1 b \\hat\{x\}=Q^\{-1\}b x^=Q−1b  
    projection matrix: P = Q Q T = I P=QQ^T=I P=QQT=I

### The Gram-Schmidt ###

*  假设  a , b , c a,b,c a,b,c三个向量不是标准正交基， 转化成正交基（基的长度不是1） A , B , C A,B,C A,B,C
 *  第一步：选 A = a A=a A=a ， B = b − A T b A T A A = e （ 误 差 ） B=b-\\frac\{A^Tb\}\{A^TA\}A=e（误差） B=b−ATAATbA=e（误差）,因为b与A的误差是与A垂直的
 *  第二步： C = c − A T c A T A A − B T c B T B B C=c-\\frac\{A^Tc\}\{A^TA\}A-\\frac\{B^Tc\}\{B^TB\}B C=c−ATAATcA−BTBBTcB，取c 在 A , B A,B A,B平面投影的误差e
 *  重复下去

例子：  
![在这里插入图片描述][20181112212638412.png]

[20181112155247602.png]: /images/20220417/9f885f3afb4043f5918cd178c43d5517.png
[2018111217192718.png]: /images/20220417/ba7baa07fe4747c29ccec2319640ff70.png
[20181112193334180.png]: /images/20220417/ae7480c241214f1586869392aa57dc58.png
[20181112192700526.png]: /images/20220417/dadf0c55d7f74e3688fe8ec7af04b5aa.png
[20181112192206447.png]: /images/20220417/9b9e665ae180489f8089816e6e696947.png
[20181112193639979.png]: /images/20220417/51ea189ec8c44b009acdd034bfa9c860.png
[- Definition]: /images/20220417/8fba5291f25a4e69a935792425139345.png
[20181112200052273.png]: /images/20220417/54e4e2c3b1c64bf2a01e9907240a3407.png
[20181112211012187.png]: /images/20220417/1e3b0def5619459a803cc9491f524d7b.png
[20181112212638412.png]: /images/20220417/80db387db741491fa85a8092c4423701.png