list manipulation - Elegant operations on matrix rows and columns

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:

2. Extracting a row from a matrix


3. Inserting a row into a matrix


4. Adding two rows within a matrix together


5. Swapping two rows


6. 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

Answer

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

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

---

Simultaneous row-and-column operations

Part is capable of working with rows and columns simultaneously⁽¹⁾.

We can operate on (or replace) a contiguous sub-array:

m2 = m;

m2[[3 ;;, 2 ;;]] /= 5;

m2 // MatrixForm

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

Or a disjoint specification:

m2 = m;

m2[[{1, 2, 4}, {1, 3}]] = 0;

m2 // MatrixForm

$\left( \begin{array}{ccc} 0 & 2 & 0 \\ 0 & 5 & 0 \\ 7 & 8 & 9 \\ 0 & 11 & 0 \\ \end{array} \right)$

Or construct a new array from constituent parts in arbitrary order:

mx = BoxMatrix[2] - 1;

mx[[{1, 2, 5, 4}, {4, 5, 1}]] = m;

mx // MatrixForm

$\left( \begin{array}{ccccc} 3 & 0 & 0 & 1 & 2 \\ 6 & 0 & 0 & 4 & 5 \\ 0 & 0 & 0 & 0 & 0 \\ 12 & 0 & 0 & 10 & 11 \\ 9 & 0 & 0 & 7 & 8 \\ \end{array} \right)$