1. 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
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:
An example of a 3x3 Orthogonal Matrix process, thus implying that A is Orthogonal.
Via Matrix Multiplication, we can see that:
For I11:
Yet I12:
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:
Where 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:
Thus, the above Symmetric, Skew-Symmetric and Orthogonal
Matrices are all Normal.