This document discusses irreducible matrix representations. It defines reducible and irreducible representations, with irreducible representations being matrices that cannot be reduced into smaller block diagonal forms through similarity transformations. The key properties of irreducible matrices are provided: 1) the sum of squares of dimensions of irreducible representations equals the group order, 2) the sum of squares of character equals the group order, and 3) characters of different representations are orthogonal. Examples of applications in quantum physics and chemistry are given related to labeling energy states based on irreducible representations.

1. IRREDUCIBLE MATRIX
PRESENTED BY
P SHASHANK REDDY
22951A66C8
CSE(AIML) B -SECTION
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PRESENTATION ON LAC

2. Types of Matrix Representation:
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• Representation is a set of matrices which
represent the operations of a point group.
It can be classified into 2 groups,
1. Reducible representation
2. Irreducible representation

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• Let us consider the C2h point group as an example. E,
C2, σh, I are the four symmetry operations present in
the group. The matrix representation for this point
group is given below.
In this case of C2h symmetry, the matrices can be
reduced to simpler matrices with smaller dimensions
(1×1 matrix).

4. Reducible Matrix
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• A matrix is reducible if and only if it can be placed into
block upper-triangular form by simultaneous row/column
permutations.
• A representation of higher dimension which can
be reduced in to representation of lower
dimension is called Reducible Representation.
• Reducible representation are called block
diagonal matrices.

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BLOCK DIAGONAL MATRICES
A block diagonal matrix is a special type of matrices, and
it has “blocks” of numbers through its “diagonal” and has
zeros elsewhere.

6. Trace of a Matrix
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The trace of a matrix (x) is the sum of its diagonal
elements.
X(A) =Σaii

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Irreducible Matrix
Because the sub-block matrices can’t be further
reduced, they are called “irreducible matrix”.
• If it is not possible to perform a similarity
transformation matrix which will reduce the matrices
of representation T, then the representation is said to
be irreducible representation.
• Generally, all 1D representations are irreducible
representations.

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• An irreducible matrix is a square and non-negative matrix such
that for every i, j there exists k > 0 such that Ak(i, j) > 0.

9. Properties of Irreducible Matrix
Rule 1: the sum of the squares of the dimensions of the
IRs(matrix) of a group is equal to the order of he group(h).
ΣAi
2 = A1
2 + A2
2 +….=h
Rule 2: the sum of the squares of the characters in any
IR = h.
𝒓 𝒙𝒊 𝒓 2 = h
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Rule 3: the vector whose components are the characters of
two different IRs are orthogonal (when multiplied with
their transpose matrix results in an identity matrix).
𝑅
𝑥 𝑖 𝑅 𝑥𝑗 𝑅 = 0

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Applications
• In quantum physics and quantum chemistry.
• Identifying the irreducible representations therefore
allows one to label the states, predict how they will
split under perturbations; or transition to other
states in V.
• Thus, in quantum mechanics, irreducible
representations of the symmetry group of the
system partially or completely label the energy
levels of the system, allowing the selection rules to
be determined.

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