What happens if the eigenvalues are complex?

What happens if the 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).

Can real eigenvalues have complex eigenvectors?

The eigenvectors are usually assumed (implicitly) to be real, but they could also be chosen as complex, it does not matter.

How do you interpret eigenvectors and eigenvalues?

Eigenvectors and Eigenvalues A right-vector is a vector as we understand them. Eigenvalues are coefficients applied to eigenvectors that give the vectors their length or magnitude. For example, a negative eigenvalue may reverse the direction of the eigenvector as part of scaling it.

How do you find complex eigenvalues?

Let A be a 2 × 2 real matrix.

1. Compute the characteristic polynomial. f ( λ )= λ 2 − Tr ( A ) λ + det ( A ) ,
2. If the eigenvalues are complex, choose one of them, and call it λ .
3. Find a corresponding (complex) eigenvalue v using the trick.
4. Then A = CBC − 1 for.

What is complex matrix?

Complex Matrices Definition. An m × n complex matrix is a rectangular array of complex numbers arranged in m rows and n columns. The set of all m × n complex matrices is denoted as. M m n C , or complex.

Can eigen values be complex?

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 interpret eigenvalues in factor analysis?

Eigenvalues represent the total amount of variance that can be explained by a given principal component. They can be positive or negative in theory, but in practice they explain variance which is always positive. If eigenvalues are greater than zero, then it’s a good sign.

What do eigenvectors tell us?

Short Answer. Eigenvectors make understanding linear transformations easy. They are the “axes” (directions) along which a linear transformation acts simply by “stretching/compressing” and/or “flipping”; eigenvalues give you the factors by which this compression occurs.

What is the difference between eigenvectors and eigenvalues?

Now let’s go back to Wikipedia’s definition of eigenvectors and eigenvalues: If T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T ( v) is a scalar multiple of v.

Does (y + IX ) V have a real eigenvalue?

Now, ( y + ix ) v is also an eigenvector of A with eigenvalue λ , as it is a scalar multiple of v . But we just showed that ( y + ix ) v is a vector with real entries, and any real eigenvector of a real matrix has a real eigenvalue. Therefore, Re ( v ) and Im ( v ) must be linearly independent after all.

What are the eigenvalues of a projection matrix?

The only eigenvalues of a projection matrix are 0 and 1. The eigenvectors for D 0 (which means Px D 0x/ fill up the nullspace. The eigenvectors for D 1 (which means Px D x/ fill up the column space. The nullspace is projected to zero. The column space projects onto itself.

What is the formula for complex eigenvalues of rotation-scaling matrices?

− i N i 1 O = N 1 − i O i N − i 1 O = N 1 i O . The most important examples of matrices with complex eigenvalues are rotation-scaling matrices, i.e., scalar multiples of rotation matrices. where a and b are real numbers, not both equal to zero. The following proposition justifies the name. be a rotation-scaling matrix. Then: