Question

\(V\) denotes a finite dimensional inner product space. Let \(T: V \rightarrow V\) be a linear operator. Show that any two of the following conditions implies the third: 1\. \(T\) is symmetric. 2\. \(T\) is an involution \(\left(T^{2}=1_{V}\right)\). 3\. \(T\) is an isometry. [Hint: In all cases, use the definition $$ \langle\mathbf{v}, T(\mathbf{w})\rangle=\langle T(\mathbf{v}), \mathbf{w}\rangle $$ of a symmetric operator. For (1) and \((3) \Rightarrow(2),\) use the fact that, if \(\left\langle T^{2}(\mathbf{v})-\mathbf{v}, \mathbf{w}\right\rangle=0\) for all \(\mathbf{w}\), then \(T^{2}(\mathbf{v})=\mathbf{v}\).]

Short answer

Any two of the conditions imply the third.

Step-by-step solution

Understand the Conditions

We need to show that any two out of three conditions - symmetry, involution, and isometry - imply the third for the linear operator \(T\). Begin by understanding what each condition means: - Symmetric: \(\langle T(\mathbf{v}), \mathbf{w}\rangle = \langle \mathbf{v}, T(\mathbf{w})\rangle\) - Involution: \(T^2 = 1_V\) (\(T\) is its own inverse).- Isometry: \(\langle T(\mathbf{v}), T(\mathbf{w})\rangle = \langle \mathbf{v}, \mathbf{w}\rangle\) for all \(\mathbf{v}, \mathbf{w} \in V\).

Symmetry and Isometry implies Involution

Assume \(T\) is symmetric and an isometry. Then, \(\langle T(\mathbf{v}), T(\mathbf{w})\rangle = \langle \mathbf{v}, \mathbf{w}\rangle\) and \(\langle T(\mathbf{v}), \mathbf{w}\rangle = \langle \mathbf{v}, T(\mathbf{w})\rangle\). Since \(T\) is an isometry, consider \(T^2\):\[\langle T^2(\mathbf{v}), \mathbf{w}\rangle = \langle T(\mathbf{v}), T(\mathbf{w})\rangle = \langle \mathbf{v}, \mathbf{w}\rangle.\]Hence, \(\langle T^2(\mathbf{v}) - \mathbf{v}, \mathbf{w}\rangle = 0\). By the given hint, this implies \(T^2(\mathbf{v}) = \mathbf{v}\) for all \(\mathbf{v} \in V\), meaning \(T\) is an involution.

Isometry and Involution implies Symmetry

Assume \(T\) is an isometry and an involution. Since \(T^2 = 1_V\), for any \(\mathbf{v}, \mathbf{w} \in V\):\[\langle T^2(\mathbf{v}), T(\mathbf{w})\rangle = \langle T(\mathbf{v}), \mathbf{w}\rangle.\]Given isometry \(\langle T(\mathbf{v}), T(\mathbf{w})\rangle = \langle \mathbf{v}, \mathbf{w}\rangle\) and knowing \(T^2(\mathbf{v}) = \mathbf{v}\), the symmetry condition follows. Therefore, \(\langle T(\mathbf{v}), \mathbf{w}\rangle = \langle \mathbf{v}, T(\mathbf{w})\rangle\), showing \(T\) is symmetric.

Symmetry and Involution implies Isometry

Assume \(T\) is symmetric and an involution, meaning \(T^2 = 1_V\) and for any \(\mathbf{v}, \mathbf{w} \in V\):\[\langle \mathbf{v}, T(\mathbf{w})\rangle = \langle T(\mathbf{v}), \mathbf{w}\rangle.\]From \(T(T(\mathbf{v})) = \mathbf{v}\), it follows:\[\langle T(\mathbf{v}), T(\mathbf{w})\rangle = \langle \mathbf{v}, \mathbf{w}\rangle.\]This confirms that \(T\) preserves the inner product; hence, \(T\) is an isometry.

Key concepts

Symmetric Operator

A symmetric operator is a linear map on an inner product space that fulfills a specific condition of symmetry in terms of its action in inner products. Mathematically, for a symmetric operator \(T\), we have:

• \(\langle T(\mathbf{v}), \mathbf{w} \rangle = \langle \mathbf{v}, T(\mathbf{w}) \rangle\)

This means that the operator commutes with the elements when taking the inner product, giving it a property similar to symmetry in matrices. This property is fundamental in many areas of mathematics and physics because it ensures that the eigenvalues are real, and eigenvectors corresponding to different eigenvalues are orthogonal. Symmetric operators often arise in contexts where the transformation describes a physical process like rotation or reflection in a manner conserving some type of geometrical property.

Involution

An involution is a linear operator \(T\) that, when applied twice, results in the identity transformation of the space it acts on. In mathematical terms, this is expressed as:

• \(T^2 = 1_V\)

This equation signifies that applying the operator \(T\) twice brings you back to the starting point, similar to how flipping a switch on and off returns it to its original state. An involution operator doesn't change any basis vectors permanently but may switch signs or reflect them temporarily. Involution is significant in other areas, such as quantum mechanics, where states can be "flipped" or "reflected". It means the operator is its own inverse, a fascinating property that intertwines with the structure of the space.

Isometry

An isometry is a transformation that preserves distances in an inner product space. Specifically, for an isometry \( T \), the condition:

• \(\langle T(\mathbf{v}), T(\mathbf{w}) \rangle = \langle \mathbf{v}, \mathbf{w} \rangle\)

This implies that the lengths of vectors and the angles between them remain unchanged under the transformation \(T\). Isometries are essential in geometry and physics, as they describe motions like rotations or translations where shapes and distances remain unaltered. These operators illustrate how certain types of transformations don't alter the "shape" of data, which is vital in applications such as computer graphics and robotics, where exact replication of forms in different orientations or positions is required.

Finite Dimensional Inner Product Space

A finite dimensional inner product space is a vector space equipped with an inner product, which is a way of multiplying vectors together to get a scalar, and the space has a finite number of dimensions. The inner product is denoted by \(\langle \mathbf{v}, \mathbf{w} \rangle\). Key features of inner product spaces include:

• An ability to define angles and lengths, allowing for notions of orthogonality and projections.
• Complete description using a basis consisting of a finite number of vectors.

These spaces form the framework for several mathematical and engineering disciplines, as they offer a geometric way to view algebraic equations. In these spaces, notions like orthogonality and norm (length of a vector) are well-defined, which facilitates solutions to complex problems in physics and signal processing through methods such as Fourier analysis and more advanced matrix decompositions.