What are the diagonal elements of a symmetric matrix?

Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.

Can a diagonal matrix have a zero on the diagonal?

Diagonal matrix is always a square matrix in which non principle diagonal elements are zero but principle diagonal elements can be zero or non zero. But Null matrix can be square or rectangular matrix. Hence we can say that square null matrix is also a diagonal matrix.

Can a symmetric matrix have complex eigenvalues?

Symmetric matrices can never have complex eigenvalues.

Can a symmetric matrix be singular?

No. The requirement for a matrix to be singular is that it have a 0 determinant.

Is diagonal matrix A zero matrix?

A diagonal matrix is one in which all non-diagonal entries are zero. Clearly this is also satisfied. Hence, a zero square matrix is upper and lower triangular as well as a diagonal matrix.

When a matrix is diagonal?

A square matrix in which every element except the principal diagonal elements is zero is called a Diagonal Matrix. A square matrix D = [dij]n x n will be called a diagonal matrix if dij = 0, whenever i is not equal to j. There are many types of matrices like the Identity matrix.

Can a matrix have both real and imaginary eigenvalues?

If each entry of an n×n matrix A is a real number, then the eigenvalues of A are all real numbers. In general, a real matrix can have a complex number eigenvalue. In fact, the part (b) gives an example of such a matrix.

Can a symmetric matrix have complex elements?

It is well known that a real symmetric matrix can be diagonalised by an orthogonal transformation. This statement is not true, in general, for a symmetric matrix of complex elements. Such complex symmetric matrices arise naturally in the study of damped vibrations of linear systems.