How many rows and columns are in an m x n matrix?

Question

A simple question: By definition, does an m x n matrix have m rows and n columns, or is it vice versa?

You can name the variables how you like though. Curiously "m by n matrix" is about twice as common as "n by m matrix" in Google search results. — Colonel Panic, Feb 18 '15 at 16:19
@ColonelPanic, that's probably because for a matrix $A$ operating on an $n$ dimensional vector $\mathcal{x}$ (i.e. $A \mathbf{x} = \mathbf{y}$) $\mathbf{y}$ is $m$ dimensional. In other words, it puts the input dimension before the output dimension alphabetically. — Shep, Apr 03 '15 at 01:42

score 49 · Accepted Answer · edited Aug 13 '15 at 21:52

49

An $m \times n$ matrix has $m$ rows and $n$ columns.

edited Aug 13 '15 at 21:52

Zhanxiong

14,040

answered Sep 06 '12 at 02:16

James

1,320

can you provide a reference/citation for this? – Anderson Green Sep 06 '12 at 02:16
3

All the textbooks i have read (both cs and math) have used this notation. For example, Strang's Introduction to Linear Algebra 4th. – James Sep 06 '12 at 02:17
You said "almost all". Were there any exceptions? – Anderson Green Sep 06 '12 at 02:19
1

Sorry, I meant all. – James Sep 06 '12 at 02:20
How come accessing elements usually start with column, e.g. in Numpy. This is confusing. – Ivan Balashov Jun 20 '16 at 06:24
1

@IvanBalashov In Numpy the first dimension is the row, not the column. – bfontaine Oct 22 '16 at 07:36
You can remember as RC – Raja Sekar Sep 17 '17 at 03:19
Does this notation convention go back to Sylvester or Cayley? – smci Mar 29 '18 at 21:16
I remember this with "mn" being in alphabetical order AND visually on the matrix "rows" are labelled first on the left, and "columns" are labelled second on top to the right, the same way we read in English, from left to right, giving the order rows x columns. – Paul Parker Sep 14 '21 at 00:59

score 1 · Answer 2 · answered Jun 04 '15 at 21:56

1

Always check and make sure you have the right convention for the occasion. Usually m x n is rows x columns. I like to remember this as being in REVERSE alphabetical order - Rows by Columns, or R first then C. However, in Boyce & DiPrima's book "Elementary Differential Equations and Boundary Value Problems" an m x n matrix has m vertical columns and n horizontal rows.
However, when addressing elements within a matrix, it's the opposite. The element "a sub i,j" references the element in the ith row and jth column. Lesson? Always check to make sure you have the correct convention!

answered Jun 04 '15 at 21:56

Sean

21

1

so much for the "universal language of mathematics" :( – Robert Lugg Jan 21 '19 at 17:32
@RobertLugg exactly. x columns by y rows. i columns by j rows. Then algebra with the 'hold my beer' goes with m rows by n columns. I almost have feelings about this... – AndrewBenjamin Apr 19 '23 at 13:15

score 1 · Answer 3 · answered Sep 06 '12 at 02:28

1

I suggest you always to check the notation on the book which you are using. I found sometimes this notation with different meaning. In advanced books, for example. Even the notation for linear maps as matrices. Sometimes they write $xT$.

answered Sep 06 '12 at 02:28

Sigur

6,416
3
25
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What does xT refer to in this case? – Anderson Green Sep 06 '12 at 02:36
It is the notation for the image of $x$ by the linear map $T$. Usually we write $T(x)$ or $Tx$. – Sigur Sep 06 '12 at 02:38

Driton Haxhiu · Answer 4 · 2019-03-13T23:21:07.250

Yes... It's m-rows and n-Columns.
Here is an example, how you can generate and read a matrix in JavaScript :)

let createMatrix = (m, n) => {
  let [row, column] = [[], []],
      rowColumn = m * n
  for (let i = 1; i <= rowColumn; i++) {
    column.push(i)
    if (i % n === 0) {
      row.push(column)
      column = []
    }
  }
  return row
}

let setColorForEachElement = (matrix, colors) => {
  let row = matrix.map(row => {
    let column = row.map((column, key) => {
      return { number: column, color: colors[key] }
    })
    return column
  })
  return row
} 

const colors = ['red', 'green', 'blue', 'purple', 'brown', 'yellow', 'orange', 'grey']
const matrix = createMatrix(6, 8)
const colorApi = setColorForEachElement(matrix, colors)

let table ='<table>'
colorApi.forEach(row => {
  table+='<tr>'
    row.forEach(column =>  table +=`<td style='background: ${column.color};'>${column.number}<td>` )
  table+='</tr>'
})

document.write(table);

score 0 · Answer 5 · answered Jan 11 '24 at 04:08

Sometimes one can forget which number maps to row or column numbers. However, we don't need remember it.

Just remember the Matrix multiplication:

For Matrix A, B, and R = A × B, we have
A (m × p)  ×  B (p × n) = R (m × n)

Now from the definition of multiplication, you will easily find that a (m × n) matrix has m rows and n columns.

How many rows and columns are in an m x n matrix?

5 Answers5