Block Diagonal Matrix and Similarity Transformation

In summary, the conversation discusses the transformation matrix, ν, and the process of diagonalization of a matrix, M. However, in some cases, matrices cannot be diagonalized and instead can only be made block-diagonal, which is known as the Jordan Form in linear algebra. The notes in the provided document may be discussing this concept.
  • #1
nigelscott
135
4
I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemist...stry-ii-fall-2008/lecture-notes/Lecture_3.pdf

How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S are the eigenvectors of M but this doesn't appear to be the case here.

Any help would be appreciated.
 
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  • #2
Some matrices cannot be diagonalized. In such cases the "best" you can do is to make them block-diagonal; in linear algebra this is called the Jordan Form. To me it looks like this is what the notes are talking about.

EDIT: right after posting, I remembered that the Jordan form has upper-triangular blocks, so the notes are not doing that...

Jason
 

FAQ: Block Diagonal Matrix and Similarity Transformation

1. What is a block diagonal matrix?

A block diagonal matrix is a square matrix that is composed of smaller square matrices along the main diagonal, with all other elements being zero. This means that the matrix is partitioned into smaller blocks, with each block representing a sub-matrix.

2. How is a block diagonal matrix different from a regular matrix?

A regular matrix has non-zero elements in every position, while a block diagonal matrix only has non-zero elements along the main diagonal. This makes a block diagonal matrix more structured and easier to work with for certain applications, such as solving systems of equations.

3. What is the purpose of a similarity transformation?

A similarity transformation is used to transform a matrix into another matrix that has the same eigenvalues and eigenvectors. This is useful in linear algebra, as it allows for simplification and easier analysis of a matrix.

4. How is a block diagonal matrix related to a similarity transformation?

A block diagonal matrix is a result of a similarity transformation, as the transformation can be used to convert a regular matrix into a block diagonal matrix. This is done by finding a matrix that has the same eigenvectors as the original matrix, and then using this matrix to transform the original matrix into a block diagonal form.

5. What are some real-world applications of block diagonal matrices and similarity transformations?

Block diagonal matrices and similarity transformations are used in a variety of fields, such as physics, engineering, and computer science. They are commonly used in solving systems of linear equations, analyzing networks and circuits, and in computer graphics and image processing. They are also important in quantum mechanics, where they are used to represent operators and observables.

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