The sum of the Eigenvalues in a given matrix is the sum of the principal diagonal elements.

Eigenvalues are a fundamental concept in linear algebra and are used to understand the behavior of linear transformations and systems of linear equations. They provide important information about the matrix and its properties.

Eigenvalues are the solutions to the characteristic equation of a matrix, which is obtained by subtracting a scalar λ from the diagonal elements of the matrix and taking the determinant. The characteristic equation is given by:

det(A - λI) = 0

where A is the matrix, λ is the scalar eigenvalue, and I is the identity matrix.

To find the eigenvalues, we solve this equation. Once we have the eigenvalues, we can compute their sum.

Now, let's understand why the sum of the Eigenvalues is the sum of the principal diagonal elements of the matrix.

Principle Diagonal Elements:
The principal diagonal elements of a matrix are the elements that lie on the diagonal from the top left to the bottom right. For example, in a 3x3 matrix, the principal diagonal elements are A[1][1], A[2][2], and A[3][3].

Eigenvalues and Principal Diagonal Elements:
When we solve the characteristic equation to find the eigenvalues, we obtain a polynomial equation. The roots of this polynomial equation are the eigenvalues of the matrix.

Each eigenvalue represents a scalar value that scales the corresponding eigenvector. Eigenvectors are the vectors that do not change their direction when a linear transformation is applied to them.

The diagonal elements of the matrix represent the scaling factors for the eigenvectors. Therefore, the sum of the eigenvalues represents the sum of the scaling factors.

The sum of the scaling factors is equal to the sum of the principal diagonal elements of the matrix.

Hence, the correct answer is option 'D': Sum of the principal diagonal elements.