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A nonsingular matrix is essentially a square matrix that holds a few special properties. It is also known as an invertible or non-degenerate matrix, primarily because of its non-zero determinant. The determinant is a key factor here as it assures that the matrix has an inverse.

This property leads to an important outcome: any system of linear equations that you construct using a nonsingular matrix will have a unique solution. In simple terms, it means there's one specific answer to the equation, which makes nonsingular matrices particularly useful in computations.

If a matrix wasn't nonsingular (i.e., if it was singular), it wouldn't have an inverse, meaning that solving equations could become much more complex or even impossible in a conventional sense.

Iterative methods are strategies for solving systems of equations by generating a succession of improving approximate solutions. One step at a time, these methods steadily get close to the exact solution.

One well-known iterative technique is the Gauss-Seidel algorithm. This method can gradually correct an initial guess until the approximation becomes sufficiently close to the precise solution.

Here's a brief overview of how it works:

• Start with an initial guess for the solution vector.
• Update the solution iteratively, recalculating elements based on current and previous values.
• Continue until the changes between successive iterations fall below a set threshold.

The success of the Gauss-Seidel method largely depends on the nature of the matrix it is applied to. For particular types of matrices, the algorithm can converge incredibly fast.
For the problem at hand, if the matrix is perfectly diagonal, Gauss-Seidel can solve it in just one step.

When we talk about a diagonal matrix, we're focusing on a unique type of matrix where all off-diagonal entries are zero. Simply put, only the elements running along the diagonal, from the top left to the bottom right corner, are non-zero.

This structure simplifies many matrix operations. Imagine you have a system described by this diagonal matrix model and need to solve it using the Gauss-Seidel algorithm. You would quickly realize that having all zeros off the diagonal means that the solution can be found immediately, without iteration.

For Gauss-Seidel, only needing one step implies that our initial guess can almost instantly be turned into an exact solution, given these diagonal matrix properties.

However, it’s crucial to note that for a diagonal matrix to remain nonsingular, none of its diagonal elements should be zero. Otherwise, the matrix would lose its invertible status, making it split into a singular form.