Elegant operations on matrix rows and columns

Question

Question

The Mathematica tutorial has a section 'Basic Matrix Operations', describing operations like transpose, inverse and determinant. These operations all work on entire matrices. I am missing a section on basic operations on matrix rows / columns.

For example:

1. Extracting a row from a matrix
2. Inserting a row into a matrix
3. Adding two rows within a matrix together
4. Swapping two rows
5. Multiplying a row with a number

And similar for columns.

What is the most elegant way to implementation of these operations? Speed is not important for me, but simplicity is.

Summary

Here I summarize my personal taste. I will update it whenever someone suggests a way I like more.

m = Range@12 ~Partition~ 3;
m // MatrixForm

$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix}$

Insert a column at position 2:

v = Range[21, 24];
Insert[m // Transpose, v, 2] // Transpose // MatrixForm

$\begin{pmatrix} 1 & 21 & 2 & 3 \\ 4 & 22 & 5 & 6 \\ 7 & 23 & 8 & 9 \\ 10 & 24& 11 & 12 \end{pmatrix}$

Extract row / column

Extract row 2:

m[[2]]

$(4,5,6)$

Extract column 2

m[[All, 2]] // MatrixForm

$\begin{pmatrix}2\\5\\8\\11\end{pmatrix}$

Insert a row / column

Insert a row at position 2:

v = Range[13, 15];
Insert[m, v, 2] // MatrixForm

$\begin{pmatrix} 1 & 2 & 3 \\ 13 & 14 & 15 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix}$

Adding two rows / columns

column 3 = column 3 + column 1:

m2 = m;  
m2[[All, 3]] += m2[[All, 1]];
m2 // MatrixForm

$\begin{pmatrix} 1 & 2 & 4 \\ 4 & 5 & 10 \\ 7 & 8 & 16 \\ 10 & 11 & 22 \end{pmatrix}$

row 2 = row 2 + row 3:

m2 = m;
m2[[2]] += m2[[3]];
m2 // MatrixForm

$\begin{pmatrix} 1 & 2 & 3 \\ 11 & 13 & 15 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix}$

Swapping rows / columns

Swap row 1 and row 3:

m2 = m;
m2[[{1, 3}]] = m2[[{3, 1}]];
m2 // MatrixForm

$\begin{pmatrix} 7 & 8 & 9 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \\ 10 & 11 & 12 \end{pmatrix}$

Swap column 1 and 3:

m2[[All, {1, 3}]] = m2[[All, {3, 1}]];
m2 // MatrixForm

$\begin{pmatrix} 3 & 2 & 1 \\ 6 & 5 & 4 \\ 9 & 8 & 7 \\ 12 & 11 & 10 \end{pmatrix}$

Multiplying rows / columns

Multiply row 2 with 2:

m*{1, 2, 1, 1} // MatrixForm

$\begin{pmatrix} 1 & 2 & 3 \\ 8 & 10 & 12 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix}$

Multiply column 1 with 5:

 ((m // Transpose)*{5, 1, 1}) // Transpose // MatrixForm

$\begin{pmatrix} 5 & 2 & 3 \\ 20 & 5 & 6 \\ 35 & 8 & 9 \\ 50 & 11 & 12 \end{pmatrix}$

References

• What is the most efficient way to add rows and columns to a matrix?
• Thanks to nikie for suggesting Matrix and Tensor Operations tutorial
• Chris Degnen pointed out http://stackoverflow.com/questions/7537401/how-to-insert-a-column-into-a-matrix-the-correct-mathematica-way

Answer 1

I like to use Part even when I don't want to modify the original matrix. This of course requires making a copy but it keeps syntax more consistent.

adding column one to column three:

m = Range@12 ~Partition~ 3;
m // MatrixForm

$\left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{array} \right)$

m2 = m;

m2[[All, 3]] += m2[[All, 1]];

m2 // MatrixForm

$\left( \begin{array}{ccc} 1 & 2 & 4 \\ 4 & 5 & 10 \\ 7 & 8 & 16 \\ 10 & 11 & 22 \end{array} \right)$

With an external vector:

v = {-1, -2, -3, -4};

m2 = m;

m2[[All, 3]] += v;

m2 // MatrixForm

$\left( \begin{array}{ccc} 1 & 2 & 2 \\ 4 & 5 & 4 \\ 7 & 8 & 6 \\ 10 & 11 & 8 \end{array} \right)$

swapping rows and columns:

m2 = m;

m2[[{1, 3}]] = m2[[{3, 1}]];

m2 // MatrixForm

$\left( \begin{array}{ccc} 7 & 8 & 9 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \\ 10 & 11 & 12 \end{array} \right)$

m2 = m;

m2[[All, {1, 3}]] = m2[[All, {3, 1}]];

m2 // MatrixForm

$\left( \begin{array}{ccc} 3 & 2 & 1 \\ 6 & 5 & 4 \\ 9 & 8 & 7 \\ 12 & 11 & 10 \end{array} \right)$

Answer 2

Interchanging rows

This'll swap rows 1 and 3.

Permute[mat, Cycles[{{1, 3}}]]

To swap columns, you can convert the permutation to a permutation list, and use

mat[[All, permList]]

Multiplying rows

This'll multiply the 3rd row by 5:

MapAt[5 # &, mat, 3]

This'll change the matrix permanently:

mat[[3]] *= 5

Answer 3

For small matrices, using simple indexing might be more readable:

Interchanging rows:

m[[{1, 3, 2}]]

Multiplying rows:

m * {1,2,1}

Adding rows

m + {0,v,0}

For large matrices, you could use SparseArray to generate the second matrix (less readable, but works for any matrix size and might be faster, too):

m * SparseArray[2 -> 2, Length[m], 1]
m + SparseArray[2 -> v, Length[m], 0]

Insert a row into a matrix

Insert[m, v, 2]

You might want to look at the Matrix and Tensor Operations tutorial, too

Answer 4

This "replace" methods work only if there are no repeated rows (or columns if you will generalize) - see comments. For more general approach see @Szabolcs solution.

m = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
m // MatrixForm

Adding rows

m /. m[[2]] -> m[[2]] + m[[3]] // MatrixForm

Interchanging rows

m /. {m[[2]] -> m[[3]], m[[3]] -> m[[2]]} // MatrixForm

Multiplying row

m /. {m[[2]] -> 3 m[[2]]} // MatrixForm

Subtracting columns

Transpose@m /. {m[[All, 2]] -> m[[All, 2]] - m[[All, 1]]} 
//Transpose // MatrixForm

Answer 5

Inserting columns (recycling answers from here).

m = Range@12~Partition~3;
m // MatrixForm
v = Range[21, 24];

MapThread[Insert, {m, v, Table[2, {Length[v]}]}] // MatrixForm

Table[Insert[m[[i]], v[[i]], 2], {i, Length[v]}] // MatrixForm

Answer 6

Not as simple as the other solutions, but the linear-algebraic treatment might be convenient in some applications:

m = Partition[Range[12], 3];

Add column 2 and column 3, and store result in column 3:

m.SparseArray[{Band[{1, 1}] -> 1, {1, 3} -> 1}, ConstantArray[Last[Dimensions[m]], 2]]

Add row 2 and row 3, and store result in row 2:

SparseArray[{Band[{1, 1}] -> 1, {2, 3} -> 1}, ConstantArray[First[Dimensions[m]], 2]].m

Multiply second row by 2:

ReplacePart[IdentityMatrix[First[Dimensions[m]]], {2, 2} -> 2].m

Multiply first column by 5:

m.ReplacePart[IdentityMatrix[Last[Dimensions[m]]], {1, 1} -> 5]