What is a Symmetric Matrix?
A square Matrix A is said to be symmetric if
for all i and j, where
is an element present at
position (
row and
column in matrix A) and
is an element present at
position (
row and
column in matrix A).
In other words, we can say that matrix A is said to be symmetric if transpose of matrix A is equal to matrix A itself (
).
Lets take an example of matrix
It is symmetric matrix because
for all i and j
Example,
,
and
In other words, transpose of Matrix A is equal to matrix A itself (
) which means matrix A is symmetric.
Lets take another example of matrix
It is not symmetric because
because
=4 and
=2.
In other words, we can say that transpose of Matrix B is not equal to matrix B (
).
What is a Skew-Symmetric Matrix?
Square Matrix A is said to be skew-symmetric if
for all i and j. In other words, we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of Matrix A i.e (
).
Note that all the main diagonal elements in skew-symmetric matrix are zero.
Lets take an example of matrix
.
It is skew-symmetric matrix because
for all i and j. Example,
= -5 and
=5 which means
. Similarly, this condition holds true for all other values of i and j.
We can also verify that Transpose of Matrix A is equal to negative of matrix A i.e
.
and
.
We can clearly see that
which makes A skew-symmetric matrix.
Lets take another example of matrix
.
Matrix B is not skew-symmetric because
or
.
Note that in skew-symmetric matrices, to make
,
,
… All the main diagonal elements have to be zero.
Adam says
Good
Abhishek says
this is good but if you can give anwer of following question .
show that the matrix A is a skew-symmetric matrix.
Al'ameen Saleh misau says
Matrix A is a skew – symmetric matrix A square matrix M is called skew symmetric if Mt=-M.