PseudoInverse
 Usage
Notes
 Further Examples
Here is a matrix.
In[1]:=
|
The inverse does not exist.
In[2]:=
|
Out[2]=
|
The pseudoinverse does exist.
In[3]:=
|
Out[3]//MatrixForm=
|
The pseudoinverse of a real matrix  , usually denoted by  , has the property that the sum of the squares of the entries of  is minimized, where  is an identity matrix of the appropriate size.
In[4]:=
|
Out[4]=
|
For invertible matrices, the pseudoinverse is the same as the inverse.
In[5]:=
|
Out[5]=
|
You can compute the pseudoinverse of a non-square matrix.
In[6]:=
|
Out[6]//MatrixForm=
|
This shows that pp satisfies the four conditions that uniquely characterize the pseudoinverse of an exact matrix.
In[7]:=
|
Out[7]=
|
In[8]:=
|
Out[8]=
|
In[9]:=
|
Out[9]=
|
In[10]:=
|
Out[10]=
|
In[11]:=
|
Using PseudoInverse to get minimal-length solutions of underdetermined systems Here is a  x  matrix and a  -vector.
In[12]:=
|
In[13]:=
|
The following system of equations is underdetermined.
In[14]:=
|
Out[14]=
|
This finds a matrix whose columns consist of independent vectors in the nullspace of mat.
In[15]:=
|
This gives the minimal-length solution.
In[16]:=
|
Out[16]=
|
In[17]:=
|
Out[17]=
|
This shows that the minimal-length solution is smaller than the original solution.
In[18]:=
|
Out[18]=
|
In[19]:=
|
| 
