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Chapter 4: Problem 14

Prove that the eigenvalues of a Hermitian matrix are real. Hint: Consider \((x, A x)\) and \(\langle A x, x\rangle .\)

Short Answer

Expert verified
The eigenvalues of a Hermitian matrix are real because \( \lambda = \lambda^* \), meaning \( \lambda \) is a real number.

Step by step solution

01

Recall the definition of a Hermitian Matrix

A Hermitian matrix is a square matrix \( A \) with complex entries that is equal to its own conjugate transpose. That is, \( A = A^* \), where \( A^* \) denotes the conjugate transpose of \( A \).
02

Define an eigenvalue problem for Hermitian Matrix

We represent the eigenvalue problem for a Hermitian matrix \( A \) as \( A x = \lambda x \), where \( \lambda \) is the eigenvalue and \( x \) is the corresponding eigenvector.
03

Consider inner product properties

Recall the properties of the inner product. In particular, for vectors \( x \) and \( y \), the inner product satisfies \( \langle x, y \rangle = \langle y, x \rangle^* \). This will be key in proving that eigenvalues are real.
04

Calculate inner product \( \langle Ax, x \rangle \)

Consider the expression \( \langle Ax, x \rangle \), we know \( Ax = \lambda x \), so this becomes \( \langle \lambda x, x \rangle = \lambda \langle x, x \rangle \).
05

Calculate inner product \( \langle x, Ax \rangle \) using Hermitian property

Using the Hermitian property, \( \langle x, Ax \rangle = \langle Ax, x \rangle^* \). So, \( \langle x, Ax \rangle = \langle \lambda x, x \rangle^* = \lambda^* \langle x, x \rangle \).
06

Equate and conclude real eigenvalue

Since \( \langle Ax, x \rangle \) must be equal to \( \langle x, Ax \rangle \) due to properties of Hermitian matrices, equating them gives \( \lambda \langle x, x \rangle = \lambda^* \langle x, x \rangle \). This simplifies to \( \lambda = \lambda^* \) since \( \langle x, x \rangle eq 0 \) (as \( x \) is an eigenvector). This implies that \( \lambda \) is a real number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eigenvalues
An eigenvalue is a special number associated with a matrix. When dealing with square matrices, eigenvalues represent the scalars that are inherent to the matrix. To find them, you take the product of the matrix and a vector (called an eigenvector) and see how the resulting vector aligns with the original. Essentially, for a matrix \( A \) and vector \( x \), the equation \( Ax = \lambda x \) is used, where \( \lambda \) is the eigenvalue.
  • In the context of a Hermitian matrix, these eigenvalues are always real numbers.
  • To determine an eigenvalue, solving the equation \( \det(A - \lambda I) = 0 \) can help, where \( I \) is the identity matrix.
The eigenvalues reveal properties about the matrix, such as whether it stretches or skews the direction of its eigenvectors. This makes eigenvalues extremely useful in fields like physics and engineering.
Inner Product
The inner product is a way to multiply two vectors in a vector space, producing a scalar. It's an extension of the dot product for complex vectors and provides a measure of the angle and length similarity between vectors. For vectors \( x \) and \( y \), the inner product is denoted by \( \langle x, y \rangle \).
  • The key property of the inner product is its symmetry: \( \langle x, y \rangle = \langle y, x \rangle^* \).
  • This concept is crucial when working with Hermitian matrices to ensure eigenvalues are real.
Through the inner product, we can also calculate lengths (or norms) and angles within vector spaces, vital insights for understanding matrix transformations.
Conjugate Transpose
To understand a Hermitian matrix, you need to grasp the concept of the conjugate transpose of a matrix. Known as the Hermitian transpose, this operation involves taking the transpose of a complex matrix and then replacing each element with its complex conjugate. For a matrix \( A \), its conjugate transpose is denoted \( A^* \).
  • The defining feature of a Hermitian matrix is that \( A = A^* \).
  • This property ensures that certain operations, like inner products, retain symmetry when substituting elements from the matrix.
Understanding the conjugate transpose is key to proving that the eigenvalues of a Hermitian matrix are real. It helps maintain the balance needed in the equality \( \langle x, Ax \rangle = \langle Ax, x \rangle^* \), crucial for preserving the property that results in real eigenvalues.

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Most popular questions from this chapter

Determine \(\operatorname{det}(A)\), where $$ A=\left[\begin{array}{lll} 2 & 1 & 0 \\ 1 & 4 & 1 \\ 0 & 1 & 2 \end{array}\right] $$ without computing the determinant by expansion by minors.

Prove that if \(A\) is nonsingular, then \(A A^{*}\) is positive definite.

Prove: If \(A\) is symmetric and nonnegative definite, then \(A=L L^{T}\) for some lower triangular matrix \(L\). The terminology nonnegative definite means that \(x^{T} A x \geq 0\) for all \(x\).

Show the resulting matrix after the forward phase of Gaussian elimination with scaled row pivoting is applied to the matrix $$ \left[\begin{array}{rrr} 2 & -2 & -4 \\ 1 & 1 & -1 \\ 3 & 7 & 5 \end{array}\right] $$ In the final matrix, write the multipliers in the appropriate locations.

Is the inequality \(\rho(A B) \leq \rho(A) \rho(B)\) true for all pairs of \(n \times n\) matrices? Is your answer the same when \(A\) and \(B\) are upper triangular?
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