This document discusses properties of symmetric, skew-symmetric, and orthogonal matrices. It defines each type of matrix and provides examples. Key points include:
- Symmetric matrices have Aij = Aji for all i and j. Skew-symmetric matrices have Aij = -Aji. Orthogonal matrices satisfy AT = A-1.
- The eigenvalues of symmetric matrices are always real. The eigenvalues of skew-symmetric matrices are either zero or purely imaginary.
- Any real square matrix can be written as the sum of a symmetric matrix and skew-symmetric matrix.
1. NPTEL โ Physics โ Mathematical Physics - 1
Lecture 16
Symmetric, Skew Symmetric and orthogonal Matrices
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1) ๐ด๐ = ๐ด
2) ๐ด๐ = โ๐ด
3) ๐ด๐ = ๐ดโ1
โถ Symmetric
โถ Skew Symmetric
โถ Orthognal
Symmetric matrix : A square matrix A is said to be symmetric if ๐ด๐๐ = ๐ด๐๐ for all i and j
where (๐, ๐)๐กโ element of the matrix denotes the intersection of ๐๐กโ and ๐๐กโ column
and similarly ๐ด๐๐ denotes the intersection of ๐๐กโ row and ๐๐กโ column of the matrix A.
Example of such a matrix is,
1 2 5
[๐ด] = [2 5 โ 7]
5 โ 7 3
Here ๐ด12 = ๐ด21 = 2; ๐ด13 = ๐ด31 = 5, ๐ด23 = ๐ด32 = โ7
Similarly an example of a skew symmetric matrix is given as
0 โ 5 4
[๐ด] = [5 0 โ 1]
โ4 1 0
Here ๐ด๐๐ = โ๐ด๐๐
It can easily be shown that ๐ด๐ = โ๐ด
Similarly the example of an orthogonal matrix is
๐ด = 2โ3
โ3 โ โ3 โ3
โ 1โ3
( 2
3
2โ3
1โ3 )
1 2 2
โ 2โ3 .
โ
One can check ๐ด๐๐ด = 1
Or ๐ด๐ = ๐ดโ1
2. NPTEL โ Physics โ Mathematical Physics - 1
Properties of different matrices
1. Every skew symmetric matrix has all the main diagonal elements zero.
๐ด๐ = โ๐ด.
๐ด + ๐ด๐ = 0. โ ๐ด๐๐ + ๐ด๐๐ = 0.
For ๐ = ๐, ๐ด๐๐ = 0
2.Any real square matrix A may be written as a sum of a symmetric matrix R and a
skew-symmetric matrix S, where,
๐ = 1
(๐ด + ๐ด๐), ๐ = 1
(๐ด โ ๐ด๐)
2 2
๐ด = ๐ + ๐.
3.Consider a square matrix A that satisfies a matrix equation of the form,
๐ด๐ฃ๐ = ๐๐ ๐ฃ๐
where ๐๐โs are called the eigenvalue and ๐ฃ๐ are eigenvector. Here ๐ฃ๐ โ 0, though ๐
๐ can be zero. Further the eigenvectors are invariant for every power of A, Or in other
words,
๐ด๐๐ฃ๐ = ๐๐
๐
๐ฃ๐
which can easily be shown as follows,
๐ด๐ฃ๐ = ๐๐๐ฃ๐
๐ด(๐ด๐ฃ๐) = ๐ด(๐๐๐ฃ๐) = ๐๐๐ด๐ฃ๐ = ๐๐
2
๐ฃ๐
The proof follows by induction.
The determination of the eigenvalues will be done shortly.
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3. Page 6 of 17
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NPTEL โ Physics โ Mathematical Physics - 1
4. The eigenvalues of a symmetric matrix are real. The eigenvalues of
a skew symmetric matrix are pure imaginary or zero.
Example The matrix [ ]
3 4
1 3
๐๐๐ก | 3 โ ๐ | = 0
1 3 โ ๐
4
(3โ๐)2 = 4.
3 โ๐ = ยฑ2 โ ๐ = 1,5 ; that is, they are real. But the matrix is not
symmetric.
So the above statement is true only in one direction. i.e. all symmetric
matrices have real eigenvalues, but all matrices with real eigenvalues
are not necessarily symmetric especially if they have same diagonal
entries.
Example
In another example, one can show that for skew symmetric matrices, the
eigenvalues are purely imaginary.
๐ด = [ 0 1 ]
โ1 0
det(๐ด โ ๐๐ผ) = 0
๐ = ยฑ๐