A simple question: By definition, does an m x n
matrix have m
rows and n
columns, or is it vice versa?
How many rows and columns are in an m x n matrix?
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0Yes it's always "{number of rows} by {number of columns}" – 2015-02-18
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2You 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. – 2015-02-18
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1@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. – 2015-04-03
4 Answers
An $m \times n$ matrix has $m$ rows and $n$ columns.
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0can you provide a reference/citation for this? – 2012-09-06
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3All the textbooks i have read (both cs and math) have used this notation. For example, Strang's Introduction to Linear Algebra 4th. – 2012-09-06
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0You said "almost all". Were there any exceptions? – 2012-09-06
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1Sorry, I meant all. – 2012-09-06
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0How come accessing elements usually start with column, e.g. in Numpy. This is confusing. – 2016-06-20
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1@IvanBalashov In Numpy the first dimension is the row, not the column. – 2016-10-22
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0You can remember as RC – 2017-09-17
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0Does this notation convention go back to [Sylvester or Cayley](https://en.wikipedia.org/wiki/Matrix_(mathematics)#History)? – 2018-03-29
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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0What does xT refer to in this case? – 2012-09-06
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0It is the notation for the image of $x$ by the linear map $T$. Usually we write $T(x)$ or $Tx$. – 2012-09-06
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!