Shilov's Linear Algebra determinant notation.

[peri]

I just started reading the first chapter of Georgi Shilov's "Linear Algebra" and I have a question about his notation for determinants. His notation, (7), for the determinant of an n x n matrix seems to be $$\det ||a_{ij}||.$$

(4) suggests Shilov would write the 1 x 1 matrix with the single element x as $$||x||.$$ So in (7), does Shilov mean for $$||a_{ij}||$$ to be interpreted as a 1 x 1 matrix or am I missing something?(4) and (7) can be found by googling for "Shilov determinant."

 

[jacobrhcp]

I think your problem is very easily solved. His notation is lousy.

By ||a_ij|| he just means the whole matrix consisting of the components a_ij, where a_ij is the single component on the ith row and jth column. This is confusing, but also conevenient if you do not want to spell out the entire matrix.

So strictly speaking, yes you're right it is a 1x1 matrix. But what he abuses the notation slightly for convenience.

 

[peri]

By ||a_ij|| he just means the whole matrix consisting of the components a_ij, where a_ij is the single component on the ith row and jth column. This is confusing, but also conevenient if you do not want to spell out the entire matrix.

That sounds like an interpretation I can live with. Thanks for taking the time to clarify his notation!

 

[RBNYC2]

I believe that "det||a_ij||" is referring to the determinant of the matrix, ||a_ij||, where i and j are variables representing the index of any row and column, respectively, and 'a' represents any element in the matrix at that row/column location. So, while a_ij can only take the value of one element at a time, this is not a matrix of just one element--he is just using one variable that can represent any element in the matrix.

This is similar to describing a set:
{x such that x is even} (for x in the set of integers)

Even though I used one variable, x, that doesn't mean my set consists of just one element, since x represents any even integer here, and thus my set contains all even integers.

 

[blue_raver22]

I am familiar with different notations used in mathematics and understand that there can be variations in notation depending on the author or textbook. In this case, it seems that Shilov's notation for determinants is slightly different from what you may have seen before.

In (7), Shilov is using the notation ||a_{ij}|| to represent the determinant of a matrix. This notation may seem unfamiliar, but it is simply a compact way of representing the matrix elements. For example, if we have a 2x2 matrix with elements a and b, Shilov's notation would be ||a b|| for the determinant.

In (4), Shilov is using the same notation for a 1x1 matrix, which is essentially a scalar value. So, in (7), ||a_{ij}|| should be interpreted as an nxn matrix, not a 1x1 matrix.

It is important to note that while Shilov's notation may differ from what you have seen before, the underlying concept and calculations for determinants remain the same. As you continue to read Shilov's book, I am sure you will become more familiar with his notation and it will become easier to understand.