## Can complex matrix have real eigenvalues?

Proof. See Datta (1995, pp. 433–439). Since a real matrix can have complex eigenvalues (occurring in complex conjugate pairs), even for a real matrix A, U and T in the above theorem can be complex.

## How do you find eigenvalues and eigenvectors of a complex matrix?

Now suppose A is a 2 × 2 matrix with a complex eigenvalue λ = a − ib, where b = 0, and corresponding eigenvector x = u + iv. That is, λ has real part a and imaginary part −b.

**How many eigenvalues does a complex matrix have?**

two eigenvalues

Since the characteristic polynomial of matrices is always a quadratic polynomial, it follows that matrices have precisely two eigenvalues — including multiplicity — and these can be described as follows.

### What happens when eigenvalues are complex?

If the n × n matrix A has real entries, its complex eigenvalues will always occur in complex conjugate pairs. This is very easy to see; recall that if an eigenvalue is complex, its eigenvectors will in general be vectors with complex entries (that is, vectors in Cn, not Rn).

### Are complex eigenvalues Diagonalizable?

In general, if a matrix has complex eigenvalues, it is not diagonalizable.

**Can a matrix with complex eigenvalues be Diagonalizable?**

#### Can eigenvalues be zero?

Eigenvalues may be equal to zero. We do not consider the zero vector to be an eigenvector: since A 0 = 0 = λ 0 for every scalar λ , the associated eigenvalue would be undefined.

#### Are complex matrices diagonalizable?

Real symmetric matrices, complex hermitian matrices, unitary matrices, and complex matrices with distinct eigenvalues are diagonalizable, i.e. conjugate to a diagonal matrix.

**How do you know if a matrix is complex diagonalizable?**

A matrix is diagonalizable if and only if for each eigenvalue the dimension of the eigenspace is equal to the multiplicity of the eigenvalue. Meaning, if you find matrices with distinct eigenvalues (multiplicity = 1) you should quickly identify those as diagonizable.

## Do all matrices have eigenvalues?

Over an algebraically closed field, every matrix has an eigenvalue. For instance, every complex matrix has an eigenvalue. Every real matrix has an eigenvalue, but it may be complex.

## What are some applications of eigenvalues and eigenvectors?

Principal Component Analysis (PCA)

**What is the eigen value of a real symmetric matrix?**

Eigenvalue of Skew Symmetric Matrix If A is a real skew-symmetric matrix then its eigenvalue will be equal to zero . Alternatively, we can say, non-zero eigenvalues of A are non-real. Every square matrix can be expressed in the form of sum of a symmetric and a skew symmetric matrix, uniquely.

### Who discovered eigenvalues of a matrix?

German Mathematician David Hilbert (1862 – 1943) is credited with naming them eigenvalues and eigenvectors.