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LinearAlgebra`MatrixManipulation`
This package includes functions for composing and separating matrices using rows, columns, and submatrices. All of the definitions involve simple combinations of built-in functions. Also included are functions for constructing a variety of special matrices.
Functions for combining matrices.
This loads the package.
In[1]:= << LinearAlgebra`MatrixManipulation`;
Define a 
matrix.
In[2]:= a = {{a11, a12}, {a21, a22}}; MatrixForm[a]
Out[2]//MatrixForm= 
Define a second matrix.
In[3]:= b = {{b11, b12}, {b21, b22}}; MatrixForm[b]
Out[3]//MatrixForm= 
This constructs a matrix by combining the columns of the two matrices.
In[4]:= AppendColumns[a, b]
Out[4]= 
Here is the same matrix displayed using MatrixForm.
In[5]:= MatrixForm[%]
Out[5]//MatrixForm= 
A matrix can also be constructed by combining the rows of these matrices.
In[6]:= AppendRows[a, b] //MatrixForm
Out[6]//MatrixForm= 
Here is a matrix constructed from submatrices a and b.
In[7]:= BlockMatrix[{{a, b}, {b, {{0, 0},
{0, 0}}}}] //MatrixForm
Out[7]//MatrixForm= 
Functions for picking out pieces of matrices.
Here is a 
matrix.
In[8]:= mat = Array[m, {3, 4}]; MatrixForm[mat]
Out[8]//MatrixForm= 
This takes the last two rows of the matrix.
In[9]:= TakeRows[mat, -2] //MatrixForm
Out[9]//MatrixForm= 
This takes the second and third columns of the matrix.
In[10]:= TakeColumns[mat, {2,3}] //MatrixForm
Out[10]//MatrixForm= 
This takes the submatrix between the element at position 
2,3 and the element at position 
3,4
.
In[11]:= TakeMatrix[mat, {2, 3}, {3, 4}] //MatrixForm
Out[11]//MatrixForm= 
Here is the same submatrix, specified as a  matrix starting with the element at position 
2,3
.
In[12]:= SubMatrix[mat, {2, 3}, {2, 2}] //MatrixForm
Out[12]//MatrixForm= 
Special matrices.
Here is an upper diagonal matrix with elements f[i,j].
In[13]:= UpperDiagonalMatrix[f, 3] //MatrixForm
Out[13]//MatrixForm= 
Matrix elements can be specified using a pure function.
In[14]:= LowerDiagonalMatrix[#1 + #2 &, 3] //MatrixForm
Out[14]//MatrixForm= 
Here is a 
Hilbert matrix.
In[15]:= HilbertMatrix[2, 4] //MatrixForm
Out[15]//MatrixForm= 
The elements of the Hankel matrix can be given as a list.
In[16]:= HankelMatrix[{w, x, y, z}] //MatrixForm
Out[16]//MatrixForm= 
Creating matrices from a set of linear equations.
This extracts the matrix of coefficients and the vector of right-hand sides from a list of linear equations.
In[17]:= LinearEquationsToMatrices[
{a[1,1]*x + a[1,2]*y == c[1],
a[2,1]*x + a[2,2]*y == c[2]}, {x, y}]
Out[17]= 
Polar decomposition of a matrix.
This computes the polar decomposition of a  matrix, then extracts the matrices  and 
.
In[18]:= {u, s} = PolarDecomposition[ {{2., 0, 0},
{3., 4., 0}, {4., 5., 6.}} ]
Out[18]= 
This gives the identity matrix.
In[19]:= u . Transpose[Conjugate[u]] // Chop
Out[19]= 
This gives the original matrix.
In[20]:= u . s // Chop
Out[20]= 
Square matrix test.
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