Special Matrices: Triangular, Symmetric, Diagonal
We have seen
that a matrix is a block of entries or two dimensional data. The size of the
matrix is given by the number of rows and the number of columns. If the two
numbers are the same, we called such matrix a square matrix.
To square matrices we associate what we call the main diagonal (in
short the diagonal). Indeed, consider the matrix
Its diagonal is given by the numbers a and d. For the matrix
its diagonal consists of a, e, and k. In general, if A
is a square matrix of order n and if aij is the number
in the ith-row and jth-colum,
then the diagonal is given by the numbers aii, for i=1,..,n.
The diagonal of a square matrix helps define two type of matrices:
upper-triangular and lower-triangular. Indeed, the diagonal
subdivides the matrix into two blocks: one above the diagonal and the other one
below it. If the lower-block consists of zeros, we call such a matrix
upper-triangular. If the upper-block consists of zeros, we call such a
matrix lower-triangular. For example, the matrices
are upper-triangular, while the matrices
are lower-triangular. Now consider the two matrices
The matrices A and B are triangular. But there is something
special about these two matrices. Indeed, as you can see if you reflect the
matrix A about the diagonal, you get the matrix B. This operation
is called the transpose operation. Indeed, let A be a nxm matrix
defined by the numbers aij, then the transpose of A,
denoted AT is the mxn matrix defined by the numbers
bij where
bij = aji.
For example, for the matrix
we have
Properties of the Transpose operation. If X and Y are
mxn matrices and Z is an nxk matrix, then
- 1.
- (X+Y)T
= XT + YT
- 2.
- (XZ)T =
ZT XT
- 3.
- (XT)T
= X
A symmetric matrix is a matrix equal to its transpose. So a symmetric
matrix must be a square matrix. For example, the matrices
are symmetric matrices. In particular a symmetric matrix of order n, contains at
most
different numbers.
A diagonal matrix is a symmetric matrix with all of its entries equal
to zero except may be the ones on the diagonal. So a diagonal matrix has at most
n different numbers. For example, the matrices
are diagonal matrices. Identity matrices are examples of diagonal matrices.
Diagonal matrices play a crucial role in matrix theory. We will see this later
on.
Example. Consider the diagonal matrix
Define the power-matrices of A by
Find the power matrices of A and then evaluate the matrices
for n=1,2,....
Answer. We have
and
By induction, one may easily show that
for every natural number n. Then we have
for n=1,2,..
Scalar Product. Consider the 3x1 matrices
The scalar product of X and Y is defined by
In particular, we have
XTX = (a2
+ b2 + c2). This is a 1 x 1 matrix .
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