How can a column vector be transformed into a diagonal matrix?

[MehranMo]

I think this is a pretty simple question. I need a transformation that will take a Column vector e.g.: <a,b,c> and turn it into a 3x3 matrix where a is in position 1,1 and b in position 2,2 and c in position 3,3. i.e.: a diagonal matrix.

Any help?

 

[Fredrik]

What kind of transformation? You can define the function by saying that for each i,j we define $X_{ij}=\delta_{ij}x_i$. (There's no summation over the repeated indices). Do you need to define the function by matrix multiplication alone, or is it OK to use addition too?

 

[I like Serena]

You could pick a 3 dimensional matrix (a 3x3x3 cube) with 1's on the main diagonal.

 

[JG89]

There is a linear isomorphism $\alpha$ such that for any vector $(a, b, c)$ $\alpha$ will take $(a,b,c)$ to the 3 by 3 matrix, whose main-diagonal entries are a, b, and c, with all other entries being 0.

 

[blue_raver22]

Yes, you are correct that this transformation can be achieved by creating a diagonal matrix. A diagonal matrix is a special type of matrix where all the elements outside the main diagonal (from top left to bottom right) are zero. In your case, the main diagonal will have the elements a, b, and c in the respective positions. This transformation is useful in many applications, such as in solving systems of linear equations or in eigenvalue calculations. To create a diagonal matrix from a column vector, you can simply use the elements of the vector as the diagonal elements of the matrix and fill the remaining elements with zeros. This transformation is straightforward and can be easily implemented in most programming languages. I hope this helps.