Function Repository Resource:

CoefficientMatrix

Source Notebook

Returns the coefficient matrix of a system of equations

Contributed by: Dennis M Schneider

ResourceFunction["CoefficientMatrix"][eqns,vars]

returns the coefficient matrix of the system of equations eqns in the variables vars.

Details and Options

The equations eqns may be a list of equations or a matrix equation.

Examples

Basic Examples (3)

Get the coefficient matrix of a linear system:

In[1]:=

$ResourceFunction[ "CoefficientMatrix"][{2 (x + y) + \[Pi] z == 3 - x, x + 2 y - z - 2 == 0}, {x, y, z}] // MatrixForm$

Out[1]=

Matrix equations are allowed:

In[2]:=

$ResourceFunction["CoefficientMatrix"][( { {3, 2, \[Pi]}, {1, 2, -1} } ) . ( { {x}, {y}, {z} } ) == ( { {a}, {b} } ), {x, y, z}] // MatrixForm$

Out[2]=

In[3]:=

$ResourceFunction["CoefficientMatrix"][( { {3, 2, \[Pi]}, {1, 2, -1} } ) . ( { {x}, {y}, {z} } ) == ( { {1, 3, 5}, {2, 4, -6} } ) . ( { {x}, {y}, {z} } ), {x, y, z}] // MatrixForm$

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Applications (5)

Find all matrices X that commute with . These matrices must satisfy the system of equations:

In[4]:=

matA = {{1, 2}, {3, 4}};
matX = Table[Subscript[x, i, j], {i, 1, 2}, {j, 1, 2}];
augmat = ResourceFunction["CoefficientMatrix"][
matA . matX == matX . matA, Flatten[matX]]

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We compute the null space of this matrix:

In[5]:=

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A basis for the matrix X:

In[6]:=

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Thus, any such matrix must be of the form:

In[7]:=

$generalmat = (\[Alpha] mat1 + \[Beta] mat2)$

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Check:

In[8]:=

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Publisher

Dennis M Schneider

Version History

1.0.0 – 31 July 2019

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License