Special sq matrices
Matrix-3
Symmetric matrix
Skew-symmetric matrix
Splitting up a sq matrix
Special sq matrices
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The part for further study may be omitted first time.
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Symmetric matrix
M is symmetric if
M = MT .
tr (sym matrix) = any number
Example :
Tr(M) = 6
1 2 8 2 2 4 8 4 3 M
4
Skew-Symmetric matrix
M is skew-symmetric if
M = - MT .
tr ( skew-sym matrix) = 0
Example :
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0 2 8 2 0 4 8 4 0 M
Symmetric matrix-1
Construction :
❶ write the diagonal with any entries ;
❷ complete 1st row with any entries ;
❸ complete 1st column with those entries used in ❷ ;
❹ complete 2nd row with any entries ;
❺ complete 2nd column with those entries used in ❹ ;
❻ complete other rows & columns using the same process.
For further study: Page- 21
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A 4×4 example:
2 / 5 4
1 7
4 0
2 6
1 2
3 5
7 6
5
2
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Skew-symmetric matrix-1
Construction :
❶ write the diagonal with any entries ;
❷ complete the 1st row with any entries ;
❸ complete the 1st column with -ve entries of ❷ ;
❹ complete the 2nd row with any entries ;
❺ complete the 2nd column with -ve entries of ❹ ;
❻ complete other rows & columns using the same process.
For further study: Page- 21
A 4x4 example:
0 2 6 9
2 0 3 7 6 3 0 0 9 7 0 0 M
Relation with square matrices
Gist of Theorem 2.7 : Gist of Theorem 2.7 :
For Further study:
Theorem 2.6
Theorem 2.7
Page 21-23 For Further study:
Theorem 2.6
Theorem 2.7
Page 21-23
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: DIU 7
Square matrix
Symmetr ic
Skew- symmetri
c
How to apply theorem 2.7
For symmetric part For skew-symmetric part
Write M
T.
Find M + M
T.
Calculate ½ ( M + M
T).
Write
M
T.
Find
M - M
T.
Calculate ½(
M - M
T ).Md Salek Parvez : Astt Prof. : GED : FSIT : DIU8
Now M = sym +
skew
Q.
Express M =as the sum of a symmetric matrix & a skew-symmetric matrix.
symmetric part skew-symmetric part
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Now M = +
Solution: Given Solution: Given
8 0
5 3 / 1
Practice Perfection ⇢
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Q-1. Write as the sum of a symm& a skew sym matrix where :
❶; ❷; ❸
Q-2. Give 3 examples of symmetric matrices: (i). 4x4 ; (ii). 5x5 ; (iii). 6x6
Q-2. Give 3 examples of symmetric matrices: (i). 4x4 ; (ii). 5x5 ; (iii). 6x6
1.Which matrices can be expressed as the sum of a symmetric &
skew-symmetric matrix?
2. What is the trace of a 29 29 matrix : (i) skew-symmetric matrix ; (ii) symmetric .
3. Give an example of 7x7 matrix which is both: symmetric and skew-symmetric matrix.
4. A 99×99 skew-symmetric matrix was found to have the trace
“0”: True or, False?
Q-3. Give 3 examples of skew- symmetric matrices: (i). 4x4 ; (ii). 5x5 ; (iii). 6x6
Q-3. Give 3 examples of skew- symmetric matrices: (i). 4x4 ; (ii). 5x5 ; (iii). 6x6
For further study (if you want to be an expert): Book Problems:
P- 79: 64-71
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