Eigenvalues are real numbers and satisfy inequality

In summary, the conversation discusses the properties of a complex unitary matrix and its relation to the eigenvalues of the matrix $A+A^{\star}$. The conversation also considers the possibility of finding the minimal polynomial of $A+A^{\star}$ and discusses the upper magnitude of 2 for the eigenvalues. There is also a question raised about the notation of $A+A^+$ and whether it was intended instead of $A+A^{\star}$.
  • #1
evinda
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Hello! (Wave)

Let $A$ be a $n \times n$ complex unitary matrix. I want to show that the eigenvalues $\lambda$ of the matrix $A+A^{\star}$ are real numbers that satisfy the relation $-2 \leq \lambda \leq 2$.

I have looked up the definitions and I read that a unitary matrix is a square matrix for which $AA^{+}=I$.

(The transpose matrix of $A^{\star}$ is symbolized with $A^{+}$.)

($A^{\star}$: complex conjugate)In order to show that the eigenvalues $\lambda$ of the matrix $A+A^{\star}$ are real numbers and satisfy that $-2 \leq \lambda \leq 2$, do we maybe have to find the minimal polynomial of the matrix $A+A^{\star}$ ? If so, how? Is there a relation? Or do we have to do it somehow else? (Thinking)
 
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  • #2
Hey evinda!

Suppose we pick $A=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$.
Then aren't the eigenvalues of $A+A^*$ imaginary? (Worried)

Can it be that $A+A^+$ was intended?

To find an upper magnitude of 2, did you consider that $\|A\mathbf x\| = \|\mathbf x\|$, which is a property of a unitary matrix? (Wondering)
 

Related to Eigenvalues are real numbers and satisfy inequality

What are eigenvalues?

Eigenvalues are a mathematical concept that represent the scalars (numbers) that are associated with a particular matrix. They are used to understand the behavior of a system or matrix, and can be calculated using various methods.

Why are eigenvalues important?

Eigenvalues are important because they provide insights into the behavior of a system or matrix. They can tell us about the stability, convergence, and other properties of a system. They are also used in many applications, such as data analysis, signal processing, and quantum mechanics.

What does it mean for eigenvalues to be real numbers?

If eigenvalues are real numbers, it means that they can be represented on the number line and do not have imaginary components. This is important because real eigenvalues can be easily visualized and understood, making them more useful in practical applications.

What does it mean for eigenvalues to satisfy an inequality?

An inequality is a mathematical statement that compares two values using symbols such as <, >, ≤, or ≥. When eigenvalues satisfy an inequality, it means that the value of the eigenvalue falls within a specific range of values. This can provide important information about the properties of a system or matrix.

How can we determine if eigenvalues satisfy an inequality?

To determine if eigenvalues satisfy an inequality, we can use various methods such as the quadratic formula or the characteristic equation. These methods involve manipulating the matrix and solving for the eigenvalues. Once we have calculated the eigenvalues, we can compare them to the given inequality to see if they satisfy it.

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