Question

Prove that if \(\mathbf{A}\) is Hermitian, then \((\mathbf{A} \mathbf{x}, \mathbf{y})=(\mathbf{x}, \mathbf{A} \mathbf{y}),\) where \(\mathbf{x}\) and \(\mathbf{y}\) are any vectors.

Short answer

Question: Prove that for a Hermitian matrix A, the inner product (Ax, y) is equal to (x, Ay) for any vectors x and y. Answer: We proved that for a Hermitian matrix A, (Ax, y) = (x, Ay) by writing the given inner products in summation notation, expanding the elements of the products Ax and Ay, changing the order of summation, and using the Hermitian property of the matrix. The order of summation does not affect the result, so we concluded that the two expressions are equal, and thus the statement is true.

Step-by-step solution

Recall the definition of a Hermitian matrix

A matrix \(\mathbf{A}\) is called Hermitian if it is equal to its conjugate transpose, i.e., \(\mathbf{A} = \mathbf{A}^H\). In other words, the elements of \(\mathbf{A}\) satisfy \(a_{ij} = \overline{a_{ji}}\) for all \(i\) and \(j\).

Write the inner product expressions

We are given two inner product expressions: \((\mathbf{Ax}, \mathbf{y})\) and \((\mathbf{x}, \mathbf{A} \mathbf{y})\). Let's write these expressions in the more familiar summation notation. The inner product in general can be written as \((\mathbf{u}, \mathbf{v}) = \sum_{i=1}^n u_i \overline{v_i}\). Using this, we can rewrite the given inner product expressions as follows: 1. \((\mathbf{Ax}, \mathbf{y}) = \sum_{i=1}^n (\mathbf{Ax})_i \overline{y_i}\) 2. \((\mathbf{x}, \mathbf{A} \mathbf{y}) = \sum_{i=1}^n x_i \overline{(\mathbf{A} \mathbf{y})_i}\)

Expand the elements of the products

Now let's expand the elements of the products \(\mathbf{Ax}\) and \(\mathbf{A} \mathbf{y}\) using the definition of matrix-vector multiplication: 1. \((\mathbf{Ax})_i = \sum_{j=1}^n a_{ij}x_j\) 2. \((\mathbf{A} \mathbf{y})_i = \sum_{j=1}^n a_{ij}y_j\) Substitute these expressions back into the inner product expressions from Step 2: 1. \((\mathbf{Ax}, \mathbf{y}) = \sum_{i=1}^n \left(\sum_{j=1}^n a_{ij}x_j\right) \overline{y_i}\) 2. \((\mathbf{x}, \mathbf{A} \mathbf{y}) = \sum_{i=1}^n x_i \overline{\left(\sum_{j=1}^n a_{ij}y_j\right)}\)

Exchange summation order and use Hermitian property

For expression (2), we can change the order of summation and use the fact that \(\mathbf{A}\) is Hermitian, which means \(a_{ij} = \overline{a_{ji}}\): \((\mathbf{x}, \mathbf{A} \mathbf{y}) = \sum_{j=1}^n \sum_{i=1}^n x_i \overline{a_{ij}}\overline{y_j} = \sum_{j=1}^n \sum_{i=1}^n x_i a_{ji} \overline{y_j}\) Notice that now the expression (2) resembles the expression (1): 1. \((\mathbf{Ax}, \mathbf{y}) = \sum_{i=1}^n \left(\sum_{j=1}^n a_{ij}x_j\right) \overline{y_i}\) 2. \((\mathbf{x}, \mathbf{A} \mathbf{y}) = \sum_{j=1}^n \sum_{i=1}^n x_i a_{ji} \overline{y_j}\)

Prove the expressions are equal

Now we can see that the two expressions are equal, as the order of summation does not affect the result of the summation. Therefore, we have proved that for a Hermitian matrix \(\mathbf{A}\): \((\mathbf{Ax}, \mathbf{y}) = (\mathbf{x}, \mathbf{A} \mathbf{y})\) This completes the proof.

Key concepts

Matrix-Vector Multiplication

Matrix-vector multiplication is a foundational concept in linear algebra. It involves multiplying a matrix by a vector, resulting in a new vector. When multiplying a matrix \( \mathbf{A} \) of size \(m \times n\) by a vector \( \mathbf{x} \) of size \(n\), the result is a new vector \( \mathbf{Ax} \) of size \(m\). Each element of the resulting vector is computed as a dot product of the rows of the matrix with the vector:

\[(\mathbf{Ax})_i = \sum_{j=1}^n a_{ij} x_j\]
This operation helps in transforming or mapping vectors from one space to another. It's particularly important in applications such as solving systems of linear equations and graphical transformations.

Inner Product Properties

The inner product is a mathematical concept that generalizes the dot product to complex vector spaces. It's particularly useful in defining concepts like length and angle in these spaces.

For vectors \( \mathbf{u} \) and \( \mathbf{v} \) in a complex space, the inner product \((\mathbf{u}, \mathbf{v})\) is defined as:

\[(\mathbf{u}, \mathbf{v}) = \sum_{i=1}^n u_i \overline{v_i}\]
Here, \( \overline{v_i} \) is the complex conjugate of \( v_i \).

• **Conjugate symmetry**: \( (\mathbf{u}, \mathbf{v}) = \overline{(\mathbf{v}, \mathbf{u})} \)
• **Linearity** in the first position: \( (a\mathbf{u} + b\mathbf{w}, \mathbf{v}) = a(\mathbf{u}, \mathbf{v}) + b(\mathbf{w}, \mathbf{v}) \)
• **Positive-definiteness**: \( (\mathbf{u}, \mathbf{u}) \geq 0 \) with equality only if \( \mathbf{u} = \mathbf{0} \)

Understanding these properties helps in analyzing symmetry and length in a complex vector space.

Conjugate Transpose

A conjugate transpose, also known as the Hermitian transpose, is an operation on a matrix that flips it over its diagonal and takes the complex conjugate of each element.

For a matrix \( \mathbf{A} \), the conjugate transpose is denoted by \( \mathbf{A}^H \). It is defined as:

\[(\mathbf{A}^H)_{ij} = \overline{(\mathbf{A})_{ji}}\]
If a matrix is equal to its own conjugate transpose, it is called Hermitian. Hermitian matrices have unique properties, such as having real eigenvalues and orthogonal eigenvectors.

Knowing how to compute and use the conjugate transpose is crucial for proofs and applications involving Hermitian matrices and understanding matrix symmetry in the complex space.

Linear Algebra Proofs

Proofs in linear algebra often require a deep understanding of core concepts and properties of matrices and vectors. For example, proving properties of Hermitian matrices involves:

• Using matrix-vector multiplication correctly.
• Applying properties of the inner product, like conjugate linearity.
• Manipulating expressions with the conjugate transpose without altering the essence, such as recognizing patterns when rearranging sums.

Proofs may also involve demonstrating equalities, like in the case of showing \((\mathbf{A} \mathbf{x}, \mathbf{y}) = (\mathbf{x}, \mathbf{A} \mathbf{y})\) for Hermitian matrices.

Each proof enhances problem-solving skills and helps reveal underlying structures and symmetries in linear algebraic systems.