- #1
nigelscott
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How does one go about finding a matrix, U, such that U-1D(g)U produces a block diagonal matrix for all g in G? For example, I am trying to figure out how the matrix (7) on page 4 of this document is obtained.
Cryo said:Finally, represent ##\mathbf{M}## in that eigenvector basis for all ##g\in G## (i.e. ##\mathbf{U}## that you wanted consists of these eigenvectors). This will be block-diagonal.
How is it given, if not already in block form, i.e. how do you know, that the two spaces are invariant?Cryo said:What bugs me is how to get block-diagonal matricies if your representation is ##\mathbf{D}_1\oplus \mathbf{D}_1##, or something similar, i.e. if you have two or more copies of the same representation. The projection operators will not touch it.
Does anyone know?
fresh_42 said:How is it given, if not already in block form, i.e. how do you know, that the two spaces are invariant?
fresh_42 said:If they are, find a basis for both and perform the change of basis on your matrices.
This is not necessarily true. It may be written in a form which is rotated away from the block form and still be a ##\mathbf{D}\oplus\mathbf{D}## representation, i.e., it may be written in a basis that mixes the irreps. This would not mean that it is not a ##\mathbf{D}\oplus\mathbf{D}## representation.fresh_42 said:I haven't checked your statements, but given they are correct, why isn't ##\mathbf{U}^\dagger \mathbf{M}\mathbf{U}## the matrix representation you are looking for? If it really equals ##\mathbf{D}\oplus \mathbf{D}## then it is in block form.
But he wrote the ##\mathbf{D}## as ##2\times 2## matrices and ##\mathbf{M}## as ##4\times 4##!Orodruin said:This is not necessarily true. It may be written in a form which is rotated away from the block form and still be a ##\mathbf{D}\oplus\mathbf{D}## representation, i.e., it may be written in a basis that mixes the irreps. This would not mean that it is not a ##\mathbf{D}\oplus\mathbf{D}## representation.
Orodruin said:This is not necessarily true. It may be written in a form which is rotated away from the block form and still be a D⊕D\mathbf{D}\oplus\mathbf{D} representation
Cryo said:How would I find ##\mathbf{U}## if I was not given it?
Can you please explain it in more detail, or maybe give a reference to it? I have 6 4-by-4 matricies, and I know that they reps of D3 and that they are isomorphic to direct sum of two copies of the two-dimensional irrep of that group. What do I do to block-diagonalize all ##\mathbf{M}##'s?fresh_42 said:look for invariants and start with all semisimple parts.
Block diagonalization is a technique used in representation theory to simplify the study of a large matrix by breaking it down into smaller, more manageable blocks. This allows for a better understanding of the structure and properties of the matrix.
Diagonalization involves finding a basis in which a matrix is represented by a diagonal matrix. Block diagonalization, on the other hand, involves finding a basis in which a matrix is represented by a block diagonal matrix, with each block representing a smaller matrix.
Block diagonalization is significant because it allows for the study of large matrices by breaking them down into smaller, more manageable blocks. This can reveal important properties and relationships between the blocks, which can then be used to understand the original matrix.
No, not all matrices can be block diagonalized. The matrix must have a certain structure, such as being symmetric or having certain symmetries, in order for block diagonalization to be possible.
Block diagonalization has various applications in fields such as physics, engineering, and computer science. It can be used to simplify and analyze large matrices in quantum mechanics, signal processing, and data compression, among others.