Question

Which of the following matrix is positive semi definite?

Answer 1

Step-by-step explanation:

A positive semidefinite matrix is a Hermitian matrix all of whose eigenvalues are nonnegative.

Here

Solving. 3rd. we get

$\left[\begin{array}{ccc}3 - \lambda&0\\0& 5 - \lambda\\\end{array}\right] \\ \\ (3 - \lambda)( 5 - \lambda) = 0 \\ \\ on \: \: solving \: \: them \: \: \: we \: get \: \: \\ \\ \lambda = 3 \: \: or \: \: 5$

Here eigenvalues are positive hence C option is positive semi definite.

A and B option gives negative eigen values and D is zero.

Answer 2

Option C is the correct answer.

• A matrix is said to be semidefinite if all the eigenvalues of the matrix are either positive (OR) zero. Any matrix having all the eigenvalues greater than or equal to zero is said to be a semidefinite matrix.
• The eigenvalues of a matrix are defined as the values of K such that the value of the determinant of the matrix ( A - KI) = 0, (where A is the matrix and I is the identity matrix).

=> | A - KI | = 0. The values K are known as Eigenvalues of the matrix.

• Consider the given options,

=> Option A:

• The eigenvalues are -3, and +5. Hence, it is not semidefinite.

=> Option B:

• The eigenvalues are 0 and +5. Hence, it is semidefinite.

=> Option C:

• The eigenvalues are -3, and -5. Hence, it is not semidefinite.

=> Option D:

• The eigenvalues are -3, and 0. Hence, it is not semidefinite.
• Only the second option satisfies the given criteria. Hence b is the correct answer.

Therefore, Option B is the correct answer.