# Eigenvalues - real and imaginary

• jpildave
In summary, if a 15x15 matrix Z is used to calculate the matrix of eigenvalues, called D, there can be two forms - diagonal or block diagonal. If D only has values on the diagonal, then there are only real values. However, if D is block diagonal, then there are both real and imaginary values. The reason for D being block diagonal is if Z is not symmetric in terms of both dimensions and values. Each entry in Z can be either a real or complex number. Additionally, diagonal is a subset of Jordan block diagonal, as one number can be counted as a Jordan block.
jpildave
Am I understanding this right?

Let's say I have a 15x15 matrix called Z. Then the matrix of eigenvalues calculated from Z, called D, can have two forms - either diagonal or block diagonal.

If the matrix D comes out with values only on the diagonal, then there are only real values. But, if the matrix D comes out block diagonal, then there are real and imaginary values.

The only reason the matrix D comes out block diagonal is if it is not symmetric, not only in terms of dimensions, but in terms of the actual values in the original matrix Z.

Correct?

I believe diagonal is a subset of Jordan block diagonal, since you can count one number as a Jordan block.

Is each entry in Z a real number or complex?

$\left[ \begin{array}{rr} 0&-1\\ 1&0\\ \end{array} \right]$
has eigenvalues = i, -i

## 1. What are eigenvalues?

Eigenvalues are a mathematical concept used in linear algebra to describe the special values of a square matrix. They represent the scalar values that, when applied to a vector, result in the same vector multiplied by a constant factor.

## 2. What is the difference between real and imaginary eigenvalues?

Real eigenvalues are values that can be expressed as a real number, while imaginary eigenvalues involve complex numbers with an imaginary component. In other words, real eigenvalues have a numerical value, while imaginary eigenvalues have a magnitude and direction in the complex plane.

## 3. How are eigenvalues calculated?

Eigenvalues are calculated by solving the characteristic equation det(A - λI) = 0, where A is the square matrix and λ is the eigenvalue. This results in a polynomial equation, and the roots of this equation are the eigenvalues of the matrix.

## 4. What is the significance of eigenvalues in science?

Eigenvalues have many applications in science, particularly in fields such as physics, engineering, and computer science. They are used to solve systems of differential equations, perform data analysis and dimensionality reduction, and model complex systems.

## 5. Can eigenvalues be complex numbers?

Yes, eigenvalues can be complex numbers. This is because complex numbers allow for a more complete representation of certain mathematical concepts, such as rotation and oscillation, which are often encountered in scientific applications.

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