Elegant operations on matrix rows and columns

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

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

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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

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

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,

permList = PermutationList[Cycles[{{1, 3}}], Last@Dimensions[mat]]

then 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

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