Chapter 6 Conditions for diagonalization

“And may it seal the door where evil dwells.” - The Great Invocation

Not all square matrices are diagonalizable. A square matrix is diagonalizable if it has enough linearly independent eigenvectors to form a complete basis for the vector space. Here are some key points to consider:

Eigenvalues and Eigenvectors: A matrix is diagonalizable if it has \(n\) linearly independent eigenvectors, where \(n\) is the size of the matrix (i.e., an \(n \times n\) matrix).
Distinct Eigenvalues: If a matrix has \(n\) distinct eigenvalues, it is guaranteed to be diagonalizable. This is because distinct eigenvalues correspond to linearly independent eigenvectors.
Defective Matrices: A matrix that does not have enough linearly independent eigenvectors is called defective and cannot be diagonalized. This often occurs when there are repeated eigenvalues and the geometric multiplicity (number of linearly independent eigenvectors) is less than the algebraic multiplicity (the number of times the eigenvalue appears).
Jordan Form: Even if a matrix is not diagonalizable, it can often be brought to a Jordan normal form, which is a block diagonal form that is as close to diagonal as possible.

In summary, while many matrices are diagonalizable, especially those with distinct eigenvalues, not all matrices have this property.

The Jordan normal form (or Jordan canonical form) is a way of representing a square matrix that may not be diagonalizable. It is a block diagonal matrix that is as close to diagonal as possible, and it provides a structured way to understand the properties of a matrix, especially when it has repeated eigenvalues.

Here are the key features of the Jordan normal form:

Jordan Blocks: The matrix is composed of one or more Jordan blocks along its diagonal. Each Jordan block corresponds to an eigenvalue of the original matrix.
Structure of Jordan Blocks: A Jordan block for an eigenvalue \(\lambda\) is a square matrix with \(\lambda\) on the diagonal, ones on the superdiagonal (the diagonal just above the main diagonal), and zeros elsewhere. For example, a Jordan block for eigenvalue \(\lambda\) of size 3 would look like this: \[ \begin{bmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{bmatrix} \]
Diagonalizability: If all Jordan blocks are \(1 \times 1\), the matrix is diagonalizable. Larger Jordan blocks indicate the presence of generalized eigenvectors and a lack of sufficient linearly independent eigenvectors.
Similarity Transformation: Any square matrix can be transformed into its Jordan normal form using a similarity transformation, which involves a change of basis in the vector space.

The Jordan normal form is particularly useful in theoretical contexts and in solving differential equations, as it simplifies the structure of matrices that are not easily diagonalizable.

6.1 Semantics

The term “diagonal” indeed has roots in Greek, where “dia” means “through” or “across,” and “gonia” means “angle.” Semantically, a diagonal is a line segment that connects two non-adjacent vertices of a polygon, effectively “cutting across” the shape.

In the context of a square or rectangle, a diagonal does bisect the shape into two equal parts, and it also bisects the right angles at the corners into two equal angles of 45 degrees each. This is because the diagonal of a square or rectangle creates two congruent right triangles, each with angles of 45, 45, and 90 degrees.

So, while “diagonal” doesn’t literally mean “two angles,” it does relate to the concept of cutting across angles or through the shape, often resulting in equal angles in the context of squares and rectangles.

6.2 Cubooctahedron (Jitterbug)

The concept of transforming squares into equilateral triangles using a 3D model like the “jitterbug” is quite fascinating. The jitterbug transformation, popularized by Buckminster Fuller, involves a dynamic geometric transformation where polyhedral shapes can morph into one another.

In this transformation, a cube (which consists of squares) can be transformed into other polyhedral shapes, such as an icosahedron, which is composed of equilateral triangles. This is achieved through a series of rotations and contractions that maintain the structural integrity of the shape while changing its form.

The jitterbug model is a great example of how geometric transformations can be visualized in three dimensions, allowing for the exploration of relationships between different polyhedral shapes. It’s a creative way to think about geometry beyond the traditional 2D plane!

The jitterbug transformation, as conceptualized by Buckminster Fuller, is more of a geometric and kinematic transformation rather than one typically described by matrix algebra. However, matrix algebra can certainly be used to model and analyze the transformations involved, especially when considering the rotations and translations of the polyhedral vertices in 3D space.

Here’s how matrix algebra might be applied:

Rotation Matrices: These are used to rotate the vertices of the polyhedra. In 3D, rotation matrices can be constructed for rotations about the x, y, or z axes. These matrices are essential for describing how the shape changes orientation during the transformation.
Transformation Matrices: These can include scaling matrices if the transformation involves changing the size of the polyhedra, as well as translation matrices if the entire shape is being moved in space.
Homogeneous Coordinates: By using homogeneous coordinates, you can combine rotation, scaling, and translation into a single transformation matrix, which simplifies the computation of complex transformations.
Sequential Transformations: The jitterbug transformation involves a sequence of movements. Each step can be represented by a matrix, and the overall transformation can be described by multiplying these matrices together.

While the jitterbug transformation is primarily a geometric concept, matrix algebra provides a powerful toolset for modeling and simulating the transformation mathematically, especially in computer graphics and computational geometry.

The “jitterbug” transformation is primarily associated with the transformation of a cuboctahedron. The cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. During the jitterbug transformation, the cuboctahedron can morph into other polyhedral shapes, such as an icosahedron or an octahedron, through a series of rotations and contractions.

This transformation showcases the dynamic relationships between these polyhedral forms, highlighting the geometric versatility and symmetry inherent in these structures. The cuboctahedron, in particular, is central to the jitterbug transformation due to its unique combination of triangular and square faces, allowing for such dynamic morphing.