`1.` A square matrix `A = [a_(ij)]` is said to be symmetric if `A-^T = A`, that is, `[a_(ij)] = [a_(ji)]` for all possible values of `i` and `j`.
For example `A= [(sqrt3,2,3),(2,-1.5,-1),(3,-1,1)]` is a symmetric matrix as `A-^T = A`
`2. ` A square matrix `A = [a_(ij)]` is said to be skew symmetric matrix if `A-^T = - A`, that is `a_(ji) = - a_(ij)` for all possible values of `i` and `j`.
Now, if we put `i = j`, we have `a_(ii) =- a_(ii)`. Therefore `2a_(ii) = 0` or `a_(ii) = 0` for all `i-'s`.
This means that all the diagonal elements of a skew symmetric matrix are zero.
For example, the matrix `B = [(0,e,f),(-e,0,g),(-f,-g,0)]` is a skew symmetric matrix as `B^T= -B`
Now, we are going to prove some results of symmetric and skew-symmetric matrices.
`text(Theorem 1.)`
For any square matrix `A` with real number entries, `A + A^T` is a symmetric matrix and `A - A^T` is a skew symmetric matrix.
Proof Let `B = A + A^T`, then
`B^T = (A + A^T)^T`
`= A^T + (A^T)^T` (as `(A + B)^T = A^T + B^T`)
`= A^T + A (as (A^T)^T = A)`
`= A + A^T` (as `A + B = B + A)`
`= B`
Therefore `B = A + A^T` is a symmetric matrix
Now let `C = A - A^T`
`C^T = (A - A′)^T = A^T - (A^T)^T ` (Why?)
`= A^T - A^T` (Why?)
`= - (A - A^T) = - C`
Therefore `C = A - A^T ` is a skew symmetric matrix.
`text(Theorem 2.)`
Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.
Proof Let `A` be a square matrix, then we can write
`A =1/2 (A+ A^T )+ 1/2 (A- A^T )`
From the Theorem 1, we know that `(A + A^T)` is a symmetric matrix and `(A - A^T)` is a skew symmetric matrix. Since for any matrix `A, (kA)^T = kA^T`, it follows that `1/2 (A+ A^T )` is symmetric matrix and `1/2 (A -A^T )` is skew symmetric matrix.
Thus, any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.
`text(Properties of Symmetric and skew-symmetric Matrices)`
(i) If A be a square matrix, then AA' and .A' A are symmetric matrices.
(ii) All positive integral powers of a symmetric matrix are symmetric,
because
(A")'= (A')"
(iii) All positive odd integral powers of a skew-symmetric matrix are
skew-symmetric and positive even integral powers of a skew-symmetric
matrix are symmetric, because
(A")' =(A')"
(iv) If A be a symmetric matrix and B be a square matrix of order that of A,
then `-A, kA, A', A^(-1) , A^n` and B' AB are also symmetric matrices, where
`n in N` and his a scalar.
(v) If A be a skew-symmetric matrix, then
(a) `A^(2n)` is a symmetric matrix for `n in N.`
(b) `A^(2n + 1)` is a skew-symmetric matrix for `n in N.`
(c) `kA` is a skew-symmetric matrix, where his scalar .
(d) B' AB is also skew-symmetric matrix, where B is a square matrix of
order that of A.
(vi) If A and Bare two symmetric matrices, then
(a) A± B, AB + BA are symmetric matrices.
(b) AB-BA is a skew-symmetric matrix
(c) AB is a symmetric matrix, iff AB = BA
vii) If A and Bare two skew-symmetric matrices, then
(a) A`ne` B, AB-BA are skew-symmetric matrices.
(b) AB + BA is a symmetric matrix.
(where A and Bare square matrices of same order)
(viii) If A be a skew-symmetric matrix and C is a column matrix, then C' AC is
a zero matrix, where C' AC is conformable.
`1.` A square matrix `A = [a_(ij)]` is said to be symmetric if `A-^T = A`, that is, `[a_(ij)] = [a_(ji)]` for all possible values of `i` and `j`.
For example `A= [(sqrt3,2,3),(2,-1.5,-1),(3,-1,1)]` is a symmetric matrix as `A-^T = A`
`2. ` A square matrix `A = [a_(ij)]` is said to be skew symmetric matrix if `A-^T = - A`, that is `a_(ji) = - a_(ij)` for all possible values of `i` and `j`.
Now, if we put `i = j`, we have `a_(ii) =- a_(ii)`. Therefore `2a_(ii) = 0` or `a_(ii) = 0` for all `i-'s`.
This means that all the diagonal elements of a skew symmetric matrix are zero.
For example, the matrix `B = [(0,e,f),(-e,0,g),(-f,-g,0)]` is a skew symmetric matrix as `B^T= -B`
Now, we are going to prove some results of symmetric and skew-symmetric matrices.
`text(Theorem 1.)`
For any square matrix `A` with real number entries, `A + A^T` is a symmetric matrix and `A - A^T` is a skew symmetric matrix.
Proof Let `B = A + A^T`, then
`B^T = (A + A^T)^T`
`= A^T + (A^T)^T` (as `(A + B)^T = A^T + B^T`)
`= A^T + A (as (A^T)^T = A)`
`= A + A^T` (as `A + B = B + A)`
`= B`
Therefore `B = A + A^T` is a symmetric matrix
Now let `C = A - A^T`
`C^T = (A - A′)^T = A^T - (A^T)^T ` (Why?)
`= A^T - A^T` (Why?)
`= - (A - A^T) = - C`
Therefore `C = A - A^T ` is a skew symmetric matrix.
`text(Theorem 2.)`
Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.
Proof Let `A` be a square matrix, then we can write
`A =1/2 (A+ A^T )+ 1/2 (A- A^T )`
From the Theorem 1, we know that `(A + A^T)` is a symmetric matrix and `(A - A^T)` is a skew symmetric matrix. Since for any matrix `A, (kA)^T = kA^T`, it follows that `1/2 (A+ A^T )` is symmetric matrix and `1/2 (A -A^T )` is skew symmetric matrix.
Thus, any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.
`text(Properties of Symmetric and skew-symmetric Matrices)`
(i) If A be a square matrix, then AA' and .A' A are symmetric matrices.
(ii) All positive integral powers of a symmetric matrix are symmetric,
because
(A")'= (A')"
(iii) All positive odd integral powers of a skew-symmetric matrix are
skew-symmetric and positive even integral powers of a skew-symmetric
matrix are symmetric, because
(A")' =(A')"
(iv) If A be a symmetric matrix and B be a square matrix of order that of A,
then `-A, kA, A', A^(-1) , A^n` and B' AB are also symmetric matrices, where
`n in N` and his a scalar.
(v) If A be a skew-symmetric matrix, then
(a) `A^(2n)` is a symmetric matrix for `n in N.`
(b) `A^(2n + 1)` is a skew-symmetric matrix for `n in N.`
(c) `kA` is a skew-symmetric matrix, where his scalar .
(d) B' AB is also skew-symmetric matrix, where B is a square matrix of
order that of A.
(vi) If A and Bare two symmetric matrices, then
(a) A± B, AB + BA are symmetric matrices.
(b) AB-BA is a skew-symmetric matrix
(c) AB is a symmetric matrix, iff AB = BA
vii) If A and Bare two skew-symmetric matrices, then
(a) A`ne` B, AB-BA are skew-symmetric matrices.
(b) AB + BA is a symmetric matrix.
(where A and Bare square matrices of same order)
(viii) If A be a skew-symmetric matrix and C is a column matrix, then C' AC is
a zero matrix, where C' AC is conformable.