Matrix Multiplication of \delta_{ij}v_j = v_i

[virgileso]

Homework Statement

Show by matrix multiplication, $$\delta_{ij}v_j = v_i$$

The Attempt at a Solution

I'm having trouble understanding how to do this, because I'm under the impression that $$v_j$$ is a row vector, which can't be multiplied by a 3x3 matrix which $$\delta_{ij}$$ is; or am I horribly wrong here?

 

[learningphysics]

I believe the Kronecker delta is just the identity matrix... if it's a 3x3 matrix, then $$v_j$$ is 3x1 (3 rows, 1 column)

 

[dextercioby]

Row vectors (n,1) can be multiplied by matrices (n,n) only if they stay at the right of the matrix, which is the case here.

 

[virgileso]

Then shouldn't the unity matrix give another row vector as an answer? I'm trying to understand how $$v_i = v_j$$

 

[dextercioby]

But the unit matrix 0 nondiagonal elements, so that v_{i}=v_{j} only for i=j.

 

[learningphysics]

Aren't we talking about column vectors here... 3x1 is a column vector... and the result of the multiplication gives the same column vector back...

$$\delta_{ij}v_j$$ denotes the sum over all j... for a particular i... ie: it is analogous to the multiplying the ith row of the matrix by the column vector $$v$$... and the result is $$v_i$$