Special type of Square Matrix.
6. Symmetric & Skew Symmetric Matrix :
Symmetric : A square matrix A = [ aij] is said to be, symmetric if,
Aij = aji " i & j" i&j
(for e.g. )
Þ (A)ij = (AT)ij Þ A = AT Þ A – AT = 0
Skew Symmetric : a square matrix A = [aij] is said to be, symmetric if,
Aij = –aji " i & j
(the pair of conjugate elements are additive inverse of each other)
Hence If A is skew symmetric , then
Aij = –aji Þ aii = 0 (putting j = i)
Aij = –(AT)ij Þ A = –AT Þ A + AT = 0
Note : (i) Max. number of distinct entries in any symmetric matrix of order n is (n(n+1)/2)
(ii) The diagonal elements of a skew square matrix are all zero, but not the converse.
(iii) For all skew symmetric matrix A of order (2n + 1) × (2n + 1), |A| = 0
Properties Of Symmetric And Skew Matrix :
P – 1 : If A be a square matrix then
(a) A + AT is symmetric matrix
(b) A – AT is skew symmetric matrix
(c) A AT and AT A are symmetric matrices Verify in each case.
(d) (AT)n = (An)T
P – 2 : Every square matrix can be uniquely expressed as the sum of a symmetric and a skew symmetric matrix.
[Solution : A= 1/2 (A+AT) + 1/2 (A-AT) = P + Q (say),
where P= 1/2 (A+AT) & Q =1/2 (A-AT)
Now, PT =(1/2 (A+AT))T = 1/2 (A+ AT)T [(kA)T = k AT]
= 1/2 (A+(AT)T) = 1/2 (A+A) = 1/2 (A+AT) Þ P is symmetric matrix
Also
Q is a skew symmetric matrix.
Thus A = P + Q when P is a symmetric matrix and Q is a skew symmetric matrix.]
Uniqueness :
If possible , Let A = R + S, where R is symmetric and S is skew symmetric. Then
AT = (R + S) T = RT + ST = R – S ( RT = R and ST = – S)
Now A = A + S and AT = R – S
Þ 1/2 ( A +AT) = P;S = (1 /2) (A-AT) =Q
Hence A is uniquely expressible as the sum of a symmetric and a skew matrix.
Examples :
1 If A and B are symmetric matrices, then show that AB is symmetric if AB = BA (i.e. A and B are commute)
[Solution : AB is symmetric
Û (AB) T = AB
Û BT AT = AB
Û BA = AB
( AT = A ; BT=B)
2. Show that the matrix BT AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
3. Show that
(a) All positive integral powers of a symmetric matrix are symmetric and
(b) All odd positive integral powers of a skew symmetric matrix are symmetric and even positive integral powers of a skew symmetric matrix are symmetric.
[Solution : Let A be a symmetric matrix and nÎN , then
(a) An = (A. A. A..........A) up to n times.
(An)T = (A. A. A..........A) T = AT. AT. ........AT up to n times.
(AT)n
(An)T = An is also a symmetric matrix.
(b) Again let A be a skew symmetric matrix.
i.e. AT = –A
Now (An)T= (AT)n (proved in (a))
(An)T = (–A) n = (–1) n (A) n
4. Let A and B be symmetric matrices of the same order then prove that
(a) A + B is a symmetric matrix.
(b) AB + BA is a symmetric matrix.
(c) AB – BA is a skew matrix.
Verify in each case
5. Express the matrix A=
as the sum of a symmetric and a skew symmetric matrix.
Solution : We have, A=
AT =
So, A + A =
and A- AT
Let P = 1/2 (A+ AT) =
and Q= 1/2 ( A- AT) =
Then PT =
and QT =
Thus P is symmetric and Q is skew symmetric. Also
P + Q =
Thus, we have expressed A as the sum of a symmetric and a skew symmetric matrix.
6. If the matrix A=
is a symmetric matrix, find x, y, z and t.
Solution : Since A = [Aij] is symmetric matrix. Therefore, aij = aij for all i, j
Þ a12 = a21, a13 = a31 and a23 = a32
Þ y = 2, x = 4 and t = –3
Thus , we have x= 4 , y = 2, t = –3 and z can assume any value.
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