# 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:

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

• you dont need All to get a row. m[[2]] and m[[2,All]] both give the second row of m. – kglr Mar 16 '12 at 8:37
• What about a partial column, say column one and first three rows, say using your example to get 1, 4, 7? I tried mat[[{1, 3}, 1]] // MatrixForm -> {1},{7}, but I want {1},{4},{7}? – sebastian c. Jan 15 '13 at 16:54
• ok got it, need to Transpose, Flatten, Take as in: Take[Flatten[Transpose[mat]], {1, 3}] -> {1,4,7}, unless there are betters way to do so? – sebastian c. Jan 15 '13 at 17:16
• How about deleting a row or column? – Hirek Feb 28 '15 at 19:00
• @Hirek, you'll want to look up Drop[] and Delete[]. – J. M.'s torpor Jun 18 '15 at 9:35

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


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

• +1. Agreed: part is just so flexible and convenient it's often the nicest way to go about these. – Szabolcs Mar 17 '12 at 7:36
• Just ask: Why won't the code m2[[All, {1, 3}]] = m2[[All, {3, 1}]]; just give column 1 and 3 indentical? First you assign column 3 to column 1, then you assign column 1 to column 3, which has already become the original column 3... – an offer can't refuse Dec 29 '15 at 11:58
• @buzhidao I don't know if I can explain this clearly for you since it seems I failed the first time. There is only a single assignment in that expression, and only a single reversal of columns. It might be translated to "let columns 1 and 3 be set to columns 3 and 1". A simpler example without columns is a = {1, 2, 3, 4, 5}; then a[[{1, 3}]] = a[[{3, 1}]]; after which a is {3, 2, 1, 4, 5}. – Mr.Wizard Dec 30 '15 at 0:10
• @buzhidao Help on WhenEvent discusses something alike: sequential actions modify variables in turn, e.g. {x[t] -> y[t], y[t] -> x[t]}, while simultaneous actions swap, e.g. {x[t], y[t]} -> {y[t], x[t]}. I think this is idiomatic to the system. – BoLe Jul 30 '16 at 9:59

### 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

• I believe one would need to take care to keep the length of the permutation list the same. As PermutationList[Cycles[{{m,n}}]] will range from 1 to Max[{m,n}]. And so mat[[All, permList]] would perhaps give a smaller matrix than intended. – Kvothe Oct 17 '18 at 9:46
• @Kvothe Thanks, updated. – Szabolcs Oct 17 '18 at 10:53

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

Interchanging rows:

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


Multiplying rows:

m * {1,2,1}


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

• For multiplying and adding rows---I agree it's likely the most efficient way. Can you show your preferred way to generate those vectors ({1,2,1} and {0,v,0}) if the matrix is large? I miss a way equivalent to mat[[2]] *= 2 which returns a copy instead of modifying the matrix. – Szabolcs Mar 16 '12 at 10:22
• @Szabolcs: Isn't m * {1,2,1} the functional equivalent to mat[[2]] *= 2? – Niki Estner Mar 16 '12 at 10:57
• If it's a 10 by 10 matrix, and you want the 7th elements, you have to write {1,1,1,1,1,1,3,1,1,1} and make sure that you inserted 3 in the correct position. This is tedious and error prone. This is why I asked how you prefer to generate that vector. – Szabolcs Mar 16 '12 at 10:59

These are ancient routines I have been using a long time ago. As a matter of fact, it's been so long that I do not even remember if I wrote them or simply shamelessly took them from some other source. Back at the time the only sources I had at my disposal where The Mathematical Journal (prior to 1998 or 1999), Bahder's wonderful book (which is the most likely source, at least of inspiration, given the style), Mathematica By Example (first edition) by Abell and Braselton and... Matlab for Engineers (LOL, I'm not kidding) by Biran and Breiner. The reason I am not sure to be the author myself is because these procedures appear too smart for me to have conceived them :-). If someone can trace the original source, I will give it due credit.

Main procedures:

row[A_,n_]:=A[[n]]
col[A_,n_]:=#[[n]]& /@ A
Col[A_,n_]:={#[[n]]}& /@ A


col returns the column in the form {x1,x2,...} Col returns it as {{x1},{x2},...} ("vertical" vector)

Smart applying:

row /: (row[A_,n_]=r_):=(A[[n]]=r)
col /: (col[A_,n_]=c_):=(A[[ Range[Dimensions[A][[1]]],{n} ]]=(List /@ c))
Col /: (Col[A_,n_]=c_):=(A[[ Range[Dimensions[A][[1]]],{n} ]]=c)
row /: (row[A_,n_]:=r_):=(A[[n]]:=r)
col /: (col[A_,n_]:=c_):=(A[[ Range[Dimensions[A][[1]]],{n} ]]:=(List /@ c))
Col /: (Col[A_,n_]:=c_):=(A[[ Range[Dimensions[A][[1]]],{n} ]]:=c)


Now... here is how to use them. Let's start with a matrix

A={
{1,2,3},
{4,5,6},
{7,8,9}
};


Suppose you want to replace the second column of A with 100 times its value. All you need to do is to tell Mathematica what is the new value of the column, for example 100 times its current value:

col[A,2]=100*col[A,2]


{200,500,800}

The side effect of col is to show the new value of the column, but its primary and intended effect is to change the original matrix A accordingly:

A


{ {1,200,3}, {4,500,6}, {7,800,9} }

row can be used in the same way. Suppose we want to substitute the first row with a linear combination of all three rows of A

row[A, 1] = row[A, 1] + 2 row[A, 2] - row[A, 3]


{2, 400, 6}

The original matrix A is changed accordingly.

A


{ {2,400,6}, {4,500,6}, {7,800,9} }

Basically these procedures allow one to do all the operations he or she wishes on rows and columns of a matrix. Extracting, defining, substituting with linear combinations or whatever comes to one's mind. As mentioned before, if one desires to extract a column in the form {{a},{b},{c}}, he should use Col instead of col.

Pretty col, uh?

EDIT: I just found a more elaborate notebook with "the making of" written by me where I refer to "Thomas Bahder's MMA for Scientists and Engineers", "Bruce Ikenaga'a Matrix Operations" and "me" as sources. So perhaps I was the author of the wrappers... Later this week I will add the procedures for joining, inserting, appending and swapping rows and columns and I will try to ascertain who wrote what.

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


• How to put this numbers in index, for example A1,A2,A3,A4..A12? m = Range@12 ~Partition~ 3; m // MatrixForm – George Mills Apr 16 '12 at 14:14
• Do you mean like this? Clear[A]; Print[ MatrixForm[m = Outer[A, {1, 2, 3, 4}, {1, 2, 3}]]]; v = Range[21, 24]; MapThread[Insert, {m, v, Table[2, {Length[v]}]}] // MatrixForm – Chris Degnen Apr 16 '12 at 18:45
• Or perhaps like this: Print[MatrixForm[m = Range@12~Partition~3]]; Clear[A]; Print[ v = Array[A, 4]]; MapThread[Insert, {m, v, Table[2, {Length[v]}]}] // MatrixForm – Chris Degnen Apr 17 '12 at 8:15
• Do we have to build Table[2, {Length[v]}]? IMO MapThread[Insert[#1, #2, 2] &, {m, v}] is cleaner. – Yi Wang Apr 15 '14 at 13:01
• It's just an alternative form. I prefer MapThread too, although now I have seen another method from Kuba: Efficient method for Inserting arrays into arrays. – Chris Degnen Apr 15 '14 at 13:07

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]


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


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


• With adding rows, I mean adding the numbers of one row to an existing row – sjdh Mar 16 '12 at 8:59
• What if the matrix has two rows that are the same? The Replace approach will affect both. – Szabolcs Mar 16 '12 at 9:09
• @sjdh I see - added an example. – Vitaliy Kaurov Mar 16 '12 at 9:09
• These ReplaceAll methods are dangerous because a matrix may contain repeated rows or columns. – Mr.Wizard Mar 16 '12 at 9:10
• @Szabolcs (and Mr.Wizard and nikie ;-) ) Very true - I'll add a comment at the top. – Vitaliy Kaurov Mar 16 '12 at 9:13

There are some internal, undocumented functions for row and columns operations:

(* in-place(!) transformation of the matrix *)
StatisticsLibraryMatrixRowTranslate[matrix, vector]
StatisticsLibraryMatrixRowTimes[matrix, vector]
StatisticsLibraryMatrixRowAffineTransform[matrix, vector, vector] (* mat, times, xlate *)
StatisticsLibraryMatrixRowAffineTransform[matrix, scalar, vector]
(* returns vector *)
StatisticsLibraryMatrixColumnSum[matrix]
StatisticsLibraryMatrixRowSum[matrix]


The first four have the inelegance, admittedly, of modifying the argument matrix in place, and so are not well-suited for functional programming. On the other hand, all six are efficient, especially on numeric arrays, packed or unpacked. Perhaps the OP's original scope does not comprise all these operations, although it is somewhat open-ended; however, I thought that since they are strongly related, it would be better to have them all here than in a separate Q&A.

Note that a vector argument can easily be a row or column of matrix, but the whole matrix will be transformed in those cases.

### Examples

mm = {
{a, b, c},
{d, e, f},
{g, h, i},
{j, k, l}
};


Add a vector to each row:

Module[{res = mm},
StatisticsLibraryMatrixRowTranslate[res, {u, v, w}];
res] // MatrixForm


Multiply each row by a vector: It's elementwise scalar multiplication, the same as row * vector, for each row in the matrix.

Module[{res = mm},
StatisticsLibraryMatrixRowTimes[res, {x, y, z}];
res] // MatrixForm


Multiply each column by a vector: Times already does what the missing MatrixColumnTimes[] would do:

mm*{w, x, y, z} // MatrixForm


Affine transformation of the rows: The MatrixRowAffineTransform can be viewed as multiplying the row by {x, y, z}] and adding the second vector {u, v, w}, or as operating on the columns threading the components of the vectors.

Module[{res = mm},
StatisticsLibraryMatrixRowAffineTransform[res, {x, y, z}, {u, v, w}];
res] // MatrixForm


MatrixRowAffineTransform allows the scaling argument to be a scalar.

Module[{res = mm},
StatisticsLibraryMatrixRowAffineTransform[res, x, {u, v, w}];
res] // MatrixForm


Affine transformation of the columns: Again, Times and Plus supply the missing MatrixColumnAffineTransform[]:

mm * {w, x, y, z} + {p, q, r, s} // MatrixForm


Summing rows or columns:

StatisticsLibraryMatrixColumnSum[mm]
(*  {a + d + g + j, b + e + h + k, c + f + i + l}  *)

StatisticsLibraryMatrixRowSum[mm]
(*  {a + b + c, d + e + f, g + h + i, j + k + l}  *)


Timings of the sums can be found here. Since the OP explicitly said speed was not an issue, I didn't want to clutter this post with timings. Nonetheless, these are, as I mentioned, fast on numeric arrays.

• (+1) In which version these functions were introduced? They aren't present in version 8.0.4. (Also you forgot to add the link in the last paragraph.) – Alexey Popkov Jul 2 '17 at 9:24
• I think in V10. I don't have access to all past versions, but they're in V10.4.1. – Michael E2 Jul 2 '17 at 17:06

As matrix multiplication is highly optimized (see here), Dot and Inner (the general form of Dot) are often very efficient methods for column manipulation.

All of the following examples somewhere make use of Dot or Inner

Matrix and vector definition

(m = Array[Subscript[a, ##] &, {2, 3}]) // MatrixForm
v = Table[Subscript[x, i], {i, 1, 3}]


$$\left( \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ \end{array} \right)$$

$$\left\{x_1,x_2,x_3\right\}$$

Multiply a Matrix by a Vector

(See this SE question: Multiplying columns with factor)

Often it is not the dot product that is of interest,

m.v // MatrixForm


$$\left\{x_1 a_{1,1}+x_2 a_{1,2}+x_3 a_{1,3},x_1 a_{2,1}+x_2 a_{2,2}+x_3 a_{2,3}\right\}$$

but

(m.DiagonalMatrix[v])//MatrixForm


$$\left( \begin{array}{ccc} x_1 a_{1,1} & x_2 a_{1,2} & x_3 a_{1,3} \\ x_1 a_{2,1} & x_2 a_{2,2} & x_3 a_{2,3} \\ \end{array} \right)$$

Inner gives the same result:

Inner[Times, m, DiagonalMatrix[v]]


In the general form, Inner[f,list1,list2,g] , 'f plays the role of multiplication and g of addition' (Inner), and the result may also be obtained with:

Inner[Times, m, v, List]


That is:

m.DiagonalMatrix[v] == Inner[Times, m, DiagonalMatrix[v]] == Inner[Times, m, v, List]


Multiply a Column by a Factor

Multiply column-4 by 10

m.DiagonalMatrix[{1, 1, 10}] // MatrixForm


$$\left( \begin{array}{ccc} a_{1,1} & a_{1,2} & 10 a_{1,3} \\ a_{2,1} & a_{2,2} & 10 a_{2,3} \\ \end{array} \right)$$

The diagonal matrix may be generated more efficiently with SparseArray. This gives the same result:

m.SparseArray[{{3, 3} -> 10, Band[{1, 1}] ->1}, Dimensions[m][[2]]]


Using Inner:

 Inner[Times, m, SparseArray[{{3, 3} -> 10, Band[{1, 1}] -> 1}, Dimensions[m][[2]]]]


or:

 Inner[Times, m, {1, 1, 10}, List]


Replace Column Entries with Zero

m.DiagonalMatrix[{1, 0, 0}] // MatrixForm


$$\left( \begin{array}{ccc} a_{1,1} & 0 & 0 \\ a_{2,1} & 0 & 0 \\ \end{array} \right)$$

Alternatively:

Inner[Times, m, {1, 0, 0}, List] // MatrixForm


m.SparseArray[{{3, 1} -> 1, Band[{1, 1}] -> 1}, Dimensions[m][[2]]] // MatrixForm


$$\left( \begin{array}{ccc} a_{1,1}+a_{1,3} & a_{1,2} & a_{1,3} \\ a_{2,1}+a_{2,3} & a_{2,2} & a_{2,3} \\ \end{array} \right)$$

For clarity:

SparseArray[{{3, 1} -> 1, Band[{1, 1}] -> 1}, Dimensions[m][[2]]] // MatrixForm


$$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right)$$

Subtract Column-3 from Column-2

m // #.SparseArray[{{3, 2} -> -1, Band[{1, 1}] -> 1}, Dimensions[#][[2]]] & // MatrixForm


$$\left( \begin{array}{ccc} a_{1,1} & a_{1,2}-a_{1,3} & a_{1,3} \\ a_{2,1} & a_{2,2}-a_{2,3} & a_{2,3} \\ \end{array} \right)$$

Subtract Column-1 from all other Columns

m // #.(SparseArray[{Band[{1, 1}] -> 1, {1, _} -> -1}, Dimensions[#][[2]]]) & // MatrixForm
`

$$\left( \begin{array}{ccc} a_{1,1} & a_{1,2}-a_{1,1} & a_{1,3}-a_{1,1} \\ a_{2,1} & a_{2,2}-a_{2,1} & a_{2,3}-a_{2,1} \\ \end{array} \right)$$