# Can a real matrix have complex eigenvalues?

## Can a real matrix have complex eigenvalues?

In general, a real matrix can have a complex number eigenvalue. In fact, the part (b) gives an example of such a matrix.

Can an invertible matrix have an eigenvalue of 0?

1) If A is not invertible then 0 is an eigenvalue. Proof of 1) Assume A is not invertible. Then Ax = 0 does not have only trivial solution by invertible matrix theorem. Since it does have the trivial solution (letting x = 0 gives a solution), but not only the trivial solution, there must be some other solution.

### Is 0 a valid eigenvalue?

Yes. 0 is an eigenvalue of a square matrix A if and only if there is a nonzero vector v with Av=0.

What does it mean if a matrix is not invertible?

A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse.

#### Why Matrix is not invertible?

We say that a square matrix is invertible if and only if the determinant is not equal to zero. In other words, a 2 x 2 matrix is only invertible if the determinant of the matrix is not 0. If the determinant is 0, then the matrix is not invertible and has no inverse.

Is a 2 invertible?

A2 is not surjective. A2 is not invertible. We clearly have Im(AB)⊂Im(A), so for B=A and with that A2 surjective means Im(A2)=ℜn.

## Is a 3×3 matrix invertible?

Not all 3×3 matrices have inverses. If the determinant of the matrix is equal to 0, then it does not have an inverse. (Notice that in the formula we divide by det(M). Division by zero is not defined.)

Is every 2×2 matrix diagonalizable?

1 Answer. Hint A matrix A with geometric multiplicity equal to its algebraic multiplicity is diagonalizable, so any nondiagonalizable 2×2 matrix must have a single eigenvalue, say, λ of algebraic multiplicity 2 but geometric multiplicity 1.

### Can every matrix be diagonalized?

In general, a rotation matrix is not diagonalizable over the reals, but all rotation matrices are diagonalizable over the complex field.

Does Diagonalizable mean invertible?

If A is diagonalizable, then A is invertible. FALSE It’s invertible if it doesn’t have zero an eigenvector but this doesn’t affect diagonalizabilty. A is diagonalizable if A has n eigenvectors.

#### Why is a matrix diagonalizable?

A square matrix is said to be diagonalizable if it is similar to a diagonal matrix. That is, A is diagonalizable if there is an invertible matrix P and a diagonal matrix D such that. A=PDP^{-1}.

Are rotation matrices Diagonalizable?

## Can a matrix be diagonalizable and not invertible?

No. For instance, the zero matrix is diagonalizable, but isn’t invertible. A square matrix is invertible if an only if its kernel is 0, and an element of the kernel is the same thing as an eigenvector with eigenvalue 0, since it is mapped to 0 times itself, which is 0.

Are orthogonal projections Diagonalizable?

Let S⊂V be a subspace of finite dimensional vector space V. Show that the orthogonal projection PS:V→S is diagonalizable.

### Can a matrix with repeated eigenvalues be Diagonalizable?

Yess, a matrix with repeated eigenvalues can be diagonalized, if the eigenspace corresponding to repeated eigenvalues has same dimension as the multiplicity of eigenvalue.

Can a non square matrix be diagonalizable?

Every matrix is not diagonalisable. Take for example non-zero nilpotent matrices. The Jordan decomposition tells us how close a given matrix can come to diagonalisability.

#### What do repeated eigenvalues mean?

We say an eigenvalue A1 of A is repeated if it is a multiple root of the char acteristic equation of A; in our case, as this is a quadratic equation, the only possible case is when A1 is a double real root. We need to find two linearly independent solutions to the system (1). We can get one solution in the usual way.

Can a symmetric matrix have repeated eigenvalues?

(i) All of the eigenvalues of a symmetric matrix are real and, hence, so are the eigenvectors. If a symmetric matrix has any repeated eigenvalues, it is still possible to determine a full set of mutually orthogonal eigenvectors, but not every full set of eigenvectors will have the orthogonality property.

## Are eigenvectors orthogonal?

In general, for any matrix, the eigenvectors are NOT always orthogonal. But for a special type of matrix, symmetric matrix, the eigenvalues are always real and the corresponding eigenvectors are always orthogonal.