Can a positive definite matrix have negative diagonal elements?

Yes. If the matrix is semi-positive definite, all the eigenvalues are non-negative.

Is a diagonal matrix always positive definite?

(c) A diagonal matrix with positive diagonal entries is positive definite. (d) A symmetric matrix with a positive determinant might not be positive definite!

Can a symmetric matrix with a positive diagonal entry be negative definite?

If a real or complex matrix is positive definite, then all of its principal minors are positive. Since the diagonal entries are the also the one-by-one principal minors of a matrix, any matrix with a diagonal entry equal to zero cannot be positive definite. (And it cannot be negative definite, either.)

Can diagonal matrix be negative?

If a matrix is strictly diagonally dominant and all its diagonal elements are positive, then the real parts of its eigenvalues are positive; if all its diagonal elements are negative, then the real parts of its eigenvalues are negative.

Are all elements of a positive definite matrix positive?

A Hermitian (or symmetric) matrix is positive definite iff all its eigenvalues are positive. Therefore, a general complex (respectively, real) matrix is positive definite iff its Hermitian (or symmetric) part has all positive eigenvalues….Positive Definite Matrix.

matrix type	OEIS	counts
(-1,0,1)-matrix	A086215	1, 7, 311, 79505.

Is a positive matrix always positive definite?

Therefore, a general complex (respectively, real) matrix is positive definite iff its Hermitian (or symmetric) part has all positive eigenvalues. The determinant of a positive definite matrix is always positive, so a positive definite matrix is always nonsingular.

Can diagonal be negative?

As we can see the diagonal entries are negative numbers and satisfy the required condition.

Can a positive definite matrix have negative eigenvalues?

We have just proved, if a matrix is positive (negative) definite, all its eigenvalues are positive (negative). If a symmetric matrix has all its eigenvalues positive (negative), it is positive (negative) definite.

What is a positive diagonal matrix?

If a matrix is strictly diagonally dominant and all its diagonal elements are positive, then the real parts of its eigenvalues are positive; if all its diagonal elements are negative, then the real parts of its eigenvalues are negative. These results follow from the Gershgorin circle theorem.