Dijkstra's Challenge Presented in honor of APL as an interactive tool turning 50 this year
Background

Ken [Iverson] presented his paper Formalism in Programming Languages in August 1963 at a Working Conference on Mechanical Language Structures, Princeton, N.J. The list of conferees is full of famous and soon-to-be famous names, and a few future Turing Award winners (Backus, Curry, Dijkstra, Floyd, Iverson, Newell, Perlis, Wilkes). The paper also records the discussion that occurred after the presentation, ending with an exchange between Ken and [Edsger] Dijkstra, in which Ken’s reply to Dijkstra’s question was a one-liner.

Challenge

How would you represent a more complex operation, for example, the sum of all elements of a matrix M which are equal to the sum of the corresponding row and column indices?

Write a snippet or expression (no need for a full program or function) to calculate the sum of each element in a given integer matrix which is equal to the sum of its indices. Or, as FryAmTheEggman puts it: given a matrix M with elements aij return the sum of each aij where aij = i + j.
You may assume the matrix already being in a variable or memory location, or you may take it as an argument or input. You may use either 0 or 1 based indices.
Test cases
 

0 for empty matrix
2

0 for 0 based indices or 2 for 1-based
1 5 2
9 4 2
5 9 6

2 for 0 based or 10 for 1 based
 0 3  0  4
 0 4  1  4
 4 3  1  2
-2 4 -2 -1

11
3 -1 3 3
3 -1 3 1

6 for 0 based or 3 for 1 based
Anecdote
Iverson's answer was ++/(M = ⍳¹ ⨢ ⍳¹)//M, which is neither valid in the Iverson Notation as defined in A Programming Language, nor in what eventually became APL. In Iverson notation, it would have been +/(M = ⍳¹(μ(M)) ⨢ ⍳¹(ν(M)))/M. In the very first versions of APL it was +/(,M=(⍳1↑⍴M)∘.+⍳1↓⍴M)/,M.
MATL, 15 14 10 bytes
3#fbb+y=*s

The input has rows separated by ;. For example: [1 5 2; 9 4 2; 5 9 6]. 1-based indexing is used.
Try it online! Or verify all test cases.
Explanation
I'll use the example with input [3 -1 3 3; 3 -1 3 1] in the explanation.
3#f    % Three-output find: for all nonzero values of implicit input matrix, pushes
       % three column vectors with row indices, column indices, and values
       %   Stack contains: [1;2;1;2;1;2;1;2], [1;1;2;2;3;3;4;4], [3;3;-1;-1;3;3;3;1]
bb     % Bubble up twice: move vectors of row and column indices to top
       %   Stack contains: [3;3;-1;-1;3;3;3;1], [1;2;1;2;1;2;1;2], [1;1;2;2;3;3;4;4]
+      % Element-wise sum of top two arrays
       %   Stack contains: [3;3;-1;-1;3;3;3;1], [2;3;3;4;4;5;5;6]
y      % Duplicate the vector of nonzero values onto the top of the stack
       %   Stack contains: [3;3;-1;-1;3;3;3;1], [2;3;3;4;4;5;5;6], [3;3;-1;-1;3;3;3;1] 
=      % Element-wise equality test of top two arrays
       %   Stack contains: [3;3;-1;-1;3;3;3;1], [0;1;0;0;0;0;0;0]
*      % Element-wise multiply of top two arrays
       %   Stack contains: [0;3;0;0;0;0;0;0]
s      % Sum of array
       %   Stack contains: 3