## What is the eigenvalue of skew-symmetric matrix?

(b) The eigenvalues of a skew-symmetric matrix are pure imaginary or zero.

**What are the eigenvalues of a symmetric matrix?**

The eigenvalues of symmetric matrices are real. Each term on the left hand side is a scalar and and since A is symmetric, the left hand side is equal to zero. But x x is the sum of products of complex numbers times their conjugates, which can never be zero unless all the numbers themselves are zero.

### What is symmetric and skew symmetric matrices?

■ A matrix is symmetric if and only if it is equal to its transpose. All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal. ■ A matrix is skew-symmetric if and only if it is the opposite of its transpose.

**Can skew-symmetric matrix have real eigenvalues?**

The eigenvalue of the skew-symmetric matrix is purely imaginary or zero.

#### What is the eigenvalue of an orthogonal matrix?

The eigenvalues of an orthogonal matrix are always ±1.

**What is meant by symmetric matrix?**

A matrix A is symmetric if it is equal to its transpose, i.e., A=AT. A matrix A is symmetric if and only if swapping indices doesn’t change its components, i.e., aij=aji.

## What is the product of eigenvalues?

The product of the n eigenvalues of A is the same as the determinant of A. If λ is an eigenvalue of A, then the dimension of Eλ is at most the multiplicity of λ. A set of eigenvectors of A, each corresponding to a different eigenvalue of A, is a linearly independent set.

**What are the properties of eigenvalues?**

Some important properties of eigen values

- Eigen values of real symmetric and hermitian matrices are real.
- Eigen values of real skew symmetric and skew hermitian matrices are either pure imaginary or zero.
- Eigen values of unitary and orthogonal matrices are of unit modulus |λ| = 1.

### Do singular matrices have eigenvalues?

Selected Properties of Eigenvalues and Eigenvectors A matrix with a 0 eigenvalue is singular, and every singular matrix has a 0 eigenvalue.