What's the Difference Between x_transpose*x and x*x_transpose?

  • Thread starter flufypancakes
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In summary, when we have a vector x, represented as (x1, x2, ..., xn)_transpose, and we multiply it by its transpose, x_transpose * x, we get x1^2 + x2^2 + ... + xn^2 = norm(x)^2. When we reverse the order and multiply x by its transpose, x * x_transpose, we get an n-by-n matrix. The asterisk in this context means "times." Matrix multiplication is defined as composing an i by j matrix with a j by k matrix to get an i by k matrix.
  • #1
flufypancakes
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0
x is a vector - (x1,x2, ..., xn)_transpose (i.e. a column vector). so when we have x_transpose*x we have x1^2+ x2^2+...+xn^2 = norm(x)^2.

right...

now what's x*x_transpose, i.e. column times the row? is it an n-by-n matrix?

actually now that I'm finished typing it I'm pretty sure it is, but i'll still post it to be sure..
 
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  • #2
What does your asterisk mean? Sometimes an asterisk means the complex conjugate of a transpose, but I'm not sure that's how you mean it here.
 
  • #3
* means "times" as in 2*2=4
 
  • #4
The definition of matrix multiplication is that when you compose an i by j matrix with a j by k matrix you get an i by k matrix.
 

Related to What's the Difference Between x_transpose*x and x*x_transpose?

1. What is the difference between X_transpose*X and X*X_transpose?

The main difference between these two expressions is the order in which the matrix operations are performed. X_transpose*X first multiplies the transpose of X with X, while X*X_transpose multiplies X with the transpose of X. This results in two different matrix products with different dimensions.

2. Which expression is more commonly used in scientific research?

The choice between X_transpose*X and X*X_transpose depends on the specific application and the desired outcome. Both expressions are commonly used in scientific research, but X_transpose*X is more often used for regression and classification problems, while X*X_transpose is used for factor analysis and principal component analysis.

3. How do the results of X_transpose*X and X*X_transpose differ?

The results of X_transpose*X and X*X_transpose will be different due to the different dimensions of the resulting matrices. X_transpose*X will result in a square matrix of dimensions (n x n), where n is the number of columns in X. On the other hand, X*X_transpose will result in a square matrix of dimensions (m x m), where m is the number of rows in X. The values within these matrices will also be different.

4. Can X_transpose*X and X*X_transpose be used interchangeably?

No, X_transpose*X and X*X_transpose cannot be used interchangeably as they produce different results. Additionally, the dimensions of the resulting matrices will also be different, making them unsuitable for the same applications.

5. Are there any advantages to using one expression over the other?

The choice between X_transpose*X and X*X_transpose depends on the specific application and the desired outcome. X_transpose*X is often preferred for regression and classification problems as it results in a matrix with the same dimensions as the original data. X*X_transpose is preferred for factor analysis and principal component analysis as it results in a matrix with interpretable principal components or factors.

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