Problem 11 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

If \(T\) is an inner product on \(V\), a linear transformation \(f: V \rightarrow V\) is called self-adjoint (with respect to \(T\) ) if \(T(x, f(y))=T(f(x), y)\) for \(x, y \in V .\) If \(v_{1}, \ldots, v_{n}\) is an orthonormal basis and \(A=\left(a_{i j}\right)\) is the matrix of \(f\) with respect to this basis, show that \(a_{i j}=a_{j i}\)

Step-by-Step Solution

Answer

For a self-adjoint linear transformation, the matrix representation is symmetric: \(a_{ij} = a_{ji}\).

Step 1: Understand the Problem

We are asked to show that if a linear transformation is self-adjoint with respect to an inner product, then the matrix representation of this transformation with respect to an orthonormal basis must be symmetric, i.e., its entries satisfy \(a_{ij} = a_{ji}\).

Step 2: Recall Self-Adjoint Property

The self-adjoint property is given by the equation\[T(x, f(y)) = T(f(x), y)\]for all \(x, y \in V\).

Step 3: Expand Inner Products

Using the given orthonormal basis \(v_{1}, \text{...}, v_{n}\), we can expand the inner products:\[T(v_i, f(v_j)) = T(f(v_i), v_j)\]where \(v_i\) and \(v_j\) are vectors from the basis.

Step 4: Express in Terms of Matrix Elements

The matrix \(A = (a_{ij})\) represents the linear transformation \(f\) with respect to the orthonormal basis. Therefore, we can write:\[f(v_j) = \sum_{k=1}^n a_{kj} v_k\]Now, we need to evaluate the inner products \(T(v_i, f(v_j))\) and \(T(f(v_i), v_j)\) using this expression.

Step 5: Calculate \(T(v_i, f(v_j))\)

We substitute the expression for \(f(v_j)\):\[T(v_i, f(v_j)) = T\left(v_i, \sum_{k=1}^n a_{kj} v_k\right)\]By linearity of the inner product, this becomes:\[T(v_i, f(v_j)) = \sum_{k=1}^n a_{kj} T(v_i, v_k)\]Using the orthonormal property \(T(v_i, v_k) = \delta_{ik}\), this simplifies to:\[T(v_i, f(v_j)) = a_{ij}\]

Step 6: Calculate \(T(f(v_i), v_j)\)

Using the self-adjoint property, we now look at the other side of the equation:\[T(f(v_i), v_j) = T\left(\sum_{l=1}^n a_{li} v_l, v_j\right)\]By linearity, this becomes:\[T(f(v_i), v_j) = \sum_{l=1}^n a_{li} T(v_l, v_j)\]Again, using the orthonormal property \(T(v_l, v_j) = \delta_{lj}\), this simplifies to:\[T(f(v_i), v_j) = a_{ji}\]

Step 7: Compare Results

We have obtained\[T(v_i, f(v_j)) = a_{ij}\]and\[T(f(v_i), v_j) = a_{ji}\]From the self-adjoint property, we know these two must be equal:\[a_{ij} = a_{ji}\]

Key Concepts

Inner Product Space

An inner product space is a vector space equipped with an inner product. The inner product is a way to multiply vectors together, leading to a scalar. Think of it as an extension of the dot product from basic vector algebra. It helps us measure angles and lengths in vector spaces.
Generally, the inner product of vectors \(x\) and \(y\) in an inner product space \(V\) is denoted by \(T(x, y)\). The inner product must satisfy these properties:

• Conjugate Symmetry: \(T(x, y) = \overline{T(y, x)}\)
• Linearity: \(T(a x + b z, y) = a T(x, y) + b T(z, y)\), for scalars \(a\) and \(b\)
• Positivity: \(T(x, x) \geq 0\), and it is zero if and only if \(x = 0\).

The inner product is fundamental for defining lengths and angles, enabling many advanced concepts in linear algebra and functional analysis.

Linear Transformation

A linear transformation is a function that maps one vector space to another (or to itself) while respecting vector addition and scalar multiplication. Formally, if \(V\) and \(W\) are vector spaces, a function \(f: V \rightarrow W\) is a linear transformation if for all vectors \(u, v \in V\) and scalars \(c\), the following hold:

• Preservation of Addition: \(f(u+v) = f(u)+f(v)\)
• Preservation of Scalar Multiplication: \(f(c \cdot u) = c \cdot f(u)\)

Linear transformations are crucial because they allow us to understand and represent various physical and geometric phenomena in a unified, linear framework. In the context of self-adjoint transformations, we explore a specific type of linear transformation that behaves specially with respect to an inner product.

Orthonormal Basis

An orthonormal basis is a set of vectors in a vector space that are both orthogonal and normalized. Orthogonal means the inner product between different vectors in the set is zero, and normalized means each vector has a length of one. Suppose \(v_{1}, v_{2}, ..., v_{n}\) form an orthonormal basis for a vector space \(V\). Then:

• \(T(v_i, v_j) = 0\) for \(i eq j\)
• \(T(v_i, v_i) = 1\) for all \(i\)

This property simplifies many computations in linear algebra. For example, projections, decompositions, and certain transformations are much easier to handle when the basis vectors are orthonormal.
In our solution, the orthonormal basis helps simplify inner products and demonstrates the symmetry of the transformation matrix directly.

Matrix Representation

A matrix representation of a linear transformation captures how the transformation acts on a vector space in a compact form. If \(f: V \rightarrow V\) is a linear transformation and \(v_1, v_2, ..., v_n\) is a basis for \(V\), we can represent \(f\) using a matrix \(A\):
\[f(v_j) = \sum_{k=1}^n a_{kj} v_k\]
Here, \(a_{kj}\) are the entries of the matrix \(A\). The entry \(a_{ij}\) tells us the component of the transformed basis vector \(f(v_j)\) along the basis vector \(v_i\).
Matrix representations let us work with linear transformations numerically and leverage standard computational tools, which is why they are so powerful in both theory and application.

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. That is, matrix \(A\) is symmetric if:
\[a_{ij} = a_{ji}\]
This means that the matrix looks the same on both sides of its main diagonal. Symmetric matrices arise naturally in many contexts, including self-adjoint (or Hermitian) transformations.
In our exercise, showing that \(a_{ij} = a_{ji}\) demonstrates that the matrix representation of a self-adjoint transformation with respect to an orthonormal basis is symmetric. This symmetry has many implications, including the fact that such matrices have real eigenvalues and can be diagonalized by an orthogonal matrix.
Understanding symmetric matrices provides deep insights into the structure and behavior of linear transformations within the inner product spaces.