Give the diagonal elements of a matrix:

Obtain the superdiagonal:

Obtain the subdiagonal:

Give a diagonal of a nonsquare matrix:

Express a matrix as the sum of its diagonal and off-diagonal parts:

The diagonal part:

Construct the diagonal as the difference between the original matrix and its diagonal part:

Confirm that two matrices have the desired properties:

Determine if the matrix is diagonalizable using its Jordan decomposition:

The superdiagonal of the Jordan form does not consist solely of zeros, so is not diagonalizable:

Confirm with a direct call to DiagonalizableMatrixQ:

Find the eigenvalues of the matrix using its Jordan decomposition:

The diagonal of the Jordan form gives the eigenvalues:

Confirm with a direct call to Eigenvalues:

For square m, DiagonalMatrix[Diagonal[m]]==m iff DiagonalMatrixQ[m] is True:

For a matrix m, Tr[m] can be expressed as a combination of Diagonal and Total:

Diagonal[m,k] for an n×n matrix gives non-empty results for 1-n<=k<=n-1:

Diagonal[m,k] gives the lowest nonzero diagonal of UpperTriangularize[m,k]:

Similarly, Diagonal[m,k] gives the highest nonzero diagonal of LowerTriangularize[m,k]:

A matrix can be reconstructed from its diagonals using Band:

For a matrix m, Diagonal[m] is equivalent to Tr[m,List]:

For a square matrix m, Diagonal[m] is equivalent to Transpose[m,{1,1}]:

Subdiagonal and superdiagonals: