A symmetric matrix is a square matrix where the elements are symmetric with respect to the main diagonal. This means that the element at row i and column j is equal to the element at row j and column i.
Real eigenvalues are the set of values that, when multiplied by a given matrix, yield a scalar multiple of the original matrix. In other words, they are the values that satisfy the equation Ax = λx, where A is the given matrix, x is a non-zero vector, and λ is the eigenvalue.
In symmetric matrices, real eigenvalues have special properties that make them useful in many applications. For example, symmetric matrices with real eigenvalues are easier to diagonalize, and their eigenvalues can provide information about the matrix's behavior and properties.
The real eigenvalues of a symmetric matrix can be found by solving the characteristic equation det(A-λI) = 0, where A is the given matrix, λ is the eigenvalue, and I is the identity matrix. This equation will result in a polynomial of degree n, where n is the size of the matrix, and the roots of this polynomial will be the real eigenvalues.
No, a symmetric matrix can only have real eigenvalues. This is because the eigenvalues of a symmetric matrix are the roots of a real polynomial, and complex roots always occur in conjugate pairs. Since a symmetric matrix has real coefficients, its eigenvalues must also be real.