The way matrices are written without boxes

[Hall]

TL;DR Summary

i-j representation of matrix.

When a matrix is represented as a box it seems all very clear, but this representation
$$
A = (a_{ij} )_{i, j =1}^{m,n}$$
Isn't very representative to me. The i -j thing creates a lot of confusion, when we write $a_{ij}$ do we mean the element of i th row and jth column or the other way round?

It becomes very ambiguous when we define the matrix multiplication and write the i-j element of AB =C as
$$
c_{ij} = \sum_{k=1}^{p} a_{ik} b_{kj}$$

It's not the case that I cannot understand it rather the matter I find it difficult to make this representation as my second nature. Can anyone of you help me in adopting this representation?

 

[PeroK]

Can anyone of you help me in adopting this representation?

It's clearly the way to go if you want to do formal proofs involving $n \times n$ matrices, where $n$ is arbitrary. It's useful, of course, to use $2 \times 2$ and $3 \times 3$ examples to help, but ultimately the compact algebraic notation is indispensible.

 

[Hall]

notation is indispensible.

Do you have something to share from your experience or your fellows experience when you were undergrads ?

 

[PeroK]

Do you have something to share from your experience or your fellows experience when you were undergrads ?

we define the matrix multiplication and write the i-j element of AB =C as
$$
c_{ij} = \sum_{k=1}^{p} a_{ik} b_{kj}$$

This IS the definition of matrix multiplication. You can't go on forever talking about taking the first row and multiplying it by the first column etc.

When I was a child, I spoke and thought and reasoned as a child. But when I grew up, I put away childish things.

Corinthians 13:11

 

[fresh_42]

For it hath been declared unto me of you, my brethren, by them which are of the house of Chloe, that there are contentions among you.

Corinthians 1:11

 

[fresh_42]

When a matrix is represented as a box it seems all very clear, but this representation
$$
A = (a_{ij} )_{i, j =1}^{m,n}$$
Isn't very representative to me.

It cannot be clearer than that! No dots indicating and so on, no guessing of the dimensions, no guessing whether the entire matrix or only a matrix entry is meant, a clear distinction between matrix and coefficients.

 

[Mark44]

When a matrix is represented as a box it seems all very clear, but this representation $$A = (a_{ij} )_{i, j =1}^{m,n}$$

Isn't very representative to me. The i -j thing creates a lot of confusion, when we write $a_{ij}$ do we mean the element of i th row and jth column or the other way round?

The notation means the matrix of elements $a_{i j}$ as i ranges from 1 to m and j ranges from 1 to n. The first index in $a_{i j}$ is the row index, and the second is the column index.

It becomes very ambiguous when we define the matrix multiplication and write the i-j element of AB =C as $c_{ij} = \sum_{k=1}^{p} a_{ik} b_{kj}$

It's not at all ambiguous once you realize that the i-j entry in matrix C is the dot product of row i in the left matrix (A) with column j in the right matrix (B). For the multiplication to be defined, or conformable, the rows of A and the columns of B must have the same number of elements, n in this case.

 

[Hall]

For it hath been declared unto me of you, my brethren, by them which are of the house of Chloe, that there are contentions among you.

Corinthians 1:11

Little children, let us not love.

John 3:18

 

[jedishrfu]

Talk no more of off topic things quoth the raven nevermore.

 

[Office_Shredder]

It becomes very ambiguous when we define the matrix multiplication and write the i-j element of AB =C as
$$
c_{ij} = \sum_{k=1}^{p} a_{ik} b_{kj}$$

You know you sum over the elements of a row of A and a column of B, so that should tell you which index is which!