A simple question: By definition, does an m x n matrix have m rows and n columns, or is it vice versa?
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asked Sep 6, 2012 at 2:15
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$\begingroup$ Yes it's always "{number of rows} by {number of columns}" $\endgroup$– Colonel PanicFeb 18, 2015 at 16:15
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2$\begingroup$ 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. $\endgroup$– Colonel PanicFeb 18, 2015 at 16:19
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2$\begingroup$ @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. $\endgroup$– ShepApr 3, 2015 at 1:42
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4 Answers
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An $m \times n$ matrix has $m$ rows and $n$ columns.
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$\begingroup$ can you provide a reference/citation for this? $\endgroup$ Sep 6, 2012 at 2:16
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3$\begingroup$ All the textbooks i have read (both cs and math) have used this notation. For example, Strang's Introduction to Linear Algebra 4th. $\endgroup$– JamesSep 6, 2012 at 2:17
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$\begingroup$ You said "almost all". Were there any exceptions? $\endgroup$ Sep 6, 2012 at 2:19
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1$\begingroup$ @IvanBalashov In Numpy the first dimension is the row, not the column. $\endgroup$ Oct 22, 2016 at 7:36
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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 $xT$.
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$\begingroup$ It is the notation for the image of $x$ by the linear map $T$. Usually we write $T(x)$ or $Tx$. $\endgroup$– SigurSep 6, 2012 at 2:38
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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!
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1$\begingroup$ so much for the "universal language of mathematics" :( $\endgroup$ Jan 21, 2019 at 17:32
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$\begingroup$ @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... $\endgroup$ Apr 19 at 13:15
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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);