lec16.ppt
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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.
![NPTEL – Physics – Mathematical Physics - 1
Lecture 16
Symmetric, Skew Symmetric and orthogonal Matrices
Page 4 of 17
Joint initiative of IITs and IISc – Funded by MHRD
1) 𝐴𝑇 = 𝐴
2) 𝐴𝑇 = −𝐴
3) 𝐴𝑇 = 𝐴−1
∶ Symmetric
∶ Skew Symmetric
∶ Orthognal
Symmetric matrix : A square matrix A is said to be symmetric if 𝐴𝑖𝑗 = 𝐴𝑗𝑖 for all i and j
where (𝑖, 𝑗)𝑡ℎ element of the matrix denotes the intersection of 𝑖𝑡ℎ and 𝑗𝑡ℎ column
and similarly 𝐴𝑗𝑖 denotes the intersection of 𝑗𝑡ℎ row and 𝑖𝑡ℎ column of the matrix A.
Example of such a matrix is,
1 2 5
[𝐴] = [2 5 − 7]
5 − 7 3
Here 𝐴12 = 𝐴21 = 2; 𝐴13 = 𝐴31 = 5, 𝐴23 = 𝐴32 = −7
Similarly an example of a skew symmetric matrix is given as
0 − 5 4
[𝐴] = [5 0 − 1]
−4 1 0
Here 𝐴𝑖𝑗 = −𝐴𝑗𝑖
It can easily be shown that 𝐴𝑇 = −𝐴
Similarly the example of an orthogonal matrix is
𝐴 = 2⁄3
⁄3 − ⁄3 ⁄3
− 1⁄3
( 2
3
2⁄3
1⁄3 )
1 2 2
− 2⁄3 .
⁄
One can check 𝐴𝑇𝐴 = 1
Or 𝐴𝑇 = 𝐴−1](https://image.slidesharecdn.com/lec16-231023100709-09fc3ae1/85/lec16-ppt-1-320.jpg)
![Page 6 of 17
Joint initiative of IITs and IISc – Funded by MHRD
NPTEL – Physics – 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
𝑑𝑒𝑡 | 3 − 𝜆 | = 0
1 3 − 𝜆
4
(3−𝜆)2 = 4.
3 −𝜆 = ±2 ⇒ 𝜆 = 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.
𝐴 = [ 0 1 ]
−1 0
det(𝐴 − 𝜆𝐼) = 0
𝜆 = ±𝑖](https://image.slidesharecdn.com/lec16-231023100709-09fc3ae1/85/lec16-ppt-3-320.jpg)
lec16.ppt
- 1. NPTEL – Physics – Mathematical Physics - 1 Lecture 16 Symmetric, Skew Symmetric and orthogonal Matrices Page 4 of 17 Joint initiative of IITs and IISc – Funded by MHRD 1) 𝐴𝑇 = 𝐴 2) 𝐴𝑇 = −𝐴 3) 𝐴𝑇 = 𝐴−1 ∶ Symmetric ∶ Skew Symmetric ∶ Orthognal Symmetric matrix : A square matrix A is said to be symmetric if 𝐴𝑖𝑗 = 𝐴𝑗𝑖 for all i and j where (𝑖, 𝑗)𝑡ℎ element of the matrix denotes the intersection of 𝑖𝑡ℎ and 𝑗𝑡ℎ column and similarly 𝐴𝑗𝑖 denotes the intersection of 𝑗𝑡ℎ row and 𝑖𝑡ℎ column of the matrix A. Example of such a matrix is, 1 2 5 [𝐴] = [2 5 − 7] 5 − 7 3 Here 𝐴12 = 𝐴21 = 2; 𝐴13 = 𝐴31 = 5, 𝐴23 = 𝐴32 = −7 Similarly an example of a skew symmetric matrix is given as 0 − 5 4 [𝐴] = [5 0 − 1] −4 1 0 Here 𝐴𝑖𝑗 = −𝐴𝑗𝑖 It can easily be shown that 𝐴𝑇 = −𝐴 Similarly the example of an orthogonal matrix is 𝐴 = 2⁄3 ⁄3 − ⁄3 ⁄3 − 1⁄3 ( 2 3 2⁄3 1⁄3 ) 1 2 2 − 2⁄3 . ⁄ One can check 𝐴𝑇𝐴 = 1 Or 𝐴𝑇 = 𝐴−1
- 2. NPTEL – Physics – Mathematical Physics - 1 Properties of different matrices 1. Every skew symmetric matrix has all the main diagonal elements zero. 𝐴𝑇 = −𝐴. 𝐴 + 𝐴𝑇 = 0. ⇒ 𝐴𝑖𝑗 + 𝐴𝑗𝑖 = 0. For 𝑗 = 𝑖, 𝐴𝑖𝑗 = 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, 𝑅 = 1 (𝐴 + 𝐴𝑇), 𝑆 = 1 (𝐴 − 𝐴𝑇) 2 2 𝐴 = 𝑅 + 𝑆. 3.Consider a square matrix A that satisfies a matrix equation of the form, 𝐴𝑣𝑖 = 𝜆𝑖 𝑣𝑖 where 𝜆𝑖’s are called the eigenvalue and 𝑣𝑖 are eigenvector. Here 𝑣𝑖 ≠ 0, though 𝜆 𝑖 can be zero. Further the eigenvectors are invariant for every power of A, Or in other words, 𝐴𝑛𝑣𝑖 = 𝜆𝑖 𝑛 𝑣𝑖 which can easily be shown as follows, 𝐴𝑣𝑖 = 𝜆𝑖𝑣𝑖 𝐴(𝐴𝑣𝑖) = 𝐴(𝜆𝑖𝑣𝑖) = 𝜆𝑖𝐴𝑣𝑖 = 𝜆𝑖 2 𝑣𝑖 The proof follows by induction. The determination of the eigenvalues will be done shortly. Page 5 of 17 Joint initiative of IITs and IISc – Funded by MHRD
- 3. Page 6 of 17 Joint initiative of IITs and IISc – Funded by MHRD NPTEL – Physics – 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 𝑑𝑒𝑡 | 3 − 𝜆 | = 0 1 3 − 𝜆 4 (3−𝜆)2 = 4. 3 −𝜆 = ±2 ⇒ 𝜆 = 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. 𝐴 = [ 0 1 ] −1 0 det(𝐴 − 𝜆𝐼) = 0 𝜆 = ±𝑖