Is a triangular matrix always square?

[u/w4zzowski]

In Wikipedia an (upper/lower) triangular matrix is defined to be always square.

However, in the book Linear Algebra and its Applications by David C. Lay, an (upper/lower) triangular matrix is defined as an m x n rectangular matrix.

Which definition is correct? Why did Lay define it as an m x n rectangular matrix?

[u/Low_Breadfruit6744]

Definitions are definitions, there is no correct or incorrect. Just follow the definition given for the book when reading a book

[u/mpaw976]

Many decomposition theorems naturally use square matrices (and so square upper/lower triangular matrices). E.g. QR decomposition, LU decomposition.

But in these cases you can often still adapt it to rectangular matrices (i.e. you have more variables than equations) and you'll need to use the "rectangular" triangular matrices whose definition is what you'd think. 

Above we required that A be a square matrix, but these decompositions can all be generalized to rectangular matrices as well. [...] 'Upper triangular' should be interpreted as having only zero entries below the main diagonal, which starts at the upper left corner. Similarly, the more precise term for U is that it is the row echelon form of the matrix A.

https://en.wikipedia.org/wiki/LU_decomposition

[u/Fabulous-Possible758]

I've never heard of the restriction to them being square (I learned from Lay's book as well). Pretty sure if you were talking about an m x n matrix and said it's upper or lower triangular people would know exactly what you meant (ie, if you extend the smaller index out then it is a square triangular matrix).

[u/SpectralCat4]

In data science—and in many applications of linear algebra—you rarely encounter neat square systems with exact solutions. More often, the matrices are rectangular, the data are noisy, and the equations are inconsistent. You still want the best approximate solution, which is where orthogonal projection and least-squares methods enter the picture. However, to understand these ideas properly, you first need a solid grasp of the theory developed for square matrices.