SPECIAL SQUARE MATRICES

Symmetric Matrices

A Matrix (A, for example) is Symmetrical if the following holds true: AT = A

In other words, the Transpose of A is identical to the Matrix A, itself. This can also be written as:

For all [i,j]: aij = aji A Matrix is a Skew-Symmetric Matrix if the following holds true:

AT = -A For all [i,j]: aij = -aji

Thus, it is clear to see that, in a Skew-Symmetric Matrix, all diagonal entries (where, in aij, i = j) must have the value of Zero, as it is implied that:

axx = -axx

axx = 0

[ 0 −2 3 12 0 4 −6

−3 −4 0 −1−1 6 1 0

] An example of a Skew-Symmetric Matrix

Orthogonal Matrices

To begin, we first re-establish the definition of Orthogonal.“Orthogonal”: Two Vectors, u and v, are said to be Orthogonal if their Dot, or Inner, Product is Zero.

A Matrix (A, for example) is Orthogonal if the following holds true: AT = A-1

In other words, the Transpose of A is identical to the Inverse of A. A different way of describing the situation is:

A AT=AT A=I n=[a b cd e fg h i ][

a d gb e hc f i ]=

1 0 00 1 00 0 1

An example of a 3x3 Orthogonal Matrix process, thus implying that A is Orthogonal.

Via Matrix Multiplication, we can see that: For I11: a2+b2+c2=1Yet I12: ad+be+cf=0

Thus, it is observed that every row of the Matrix A is Orthogonal with every other row of A (but not with itself). Furthermore, each row can also be seen as Unit Vectors, and thus described as u1, u2, u3.

Generalising this case, we can say that, if {u1, u2, …, un} are all Unit Vectors and Orthogonal to each other, {u1, u2, …, un} in Rn form an Orthonormal Set. This can be displayed as:

δ ij=ui ∙ u j={1: i= j0 : i≠ j

Where δ ij is the Kronecker Delta Function, i.e. the function that returns 1 if all its arguments are equal, and 0 if they are not.

Normal Matrices

A Matrix is said to be Normal if it commutes with its transpose. In other words, the following must be true:

AAT=AT A

Thus, the above Symmetric, Skew-Symmetric and Orthogonal Matrices are all Normal.