Special Matrices and Definitions
SQUARE MATRIX
A square matrix is one that has the sane number of rows and columns; i.e., \(n \times n\).
\[
\text{e.g.} \quad A = \begin{bmatrix}
5 & -1 & 9 \\
2 & 3 & 8\\
-2 & 0 & 1 \\
\end{bmatrix}
\]
MAIN DIAGONAL
For a square matrix, the main diagonal is the diagonal that starts at the top left of the matrix
and goes down to the bottom left of the matrix.
\[
\begin{bmatrix}
\colorbox{yellow}{$a_{11}$} & a_{12} & a_{13} \\
a_{21}& \colorbox{yellow}{$a_{22}$} & a_{23}\\
a_{31} & a_{32} & \colorbox{yellow}{$a_{33}$} \\
\end{bmatrix}\]
DIAGONAL MATRIX
A square matrix that has entries along the main diagonal and zeroes everywhere else.
\[
\text{e.g.} \quad A = \begin{bmatrix}
4 & 0 & 0 \\
0& 1 & 0\\
0 & 0 & -9 \\
\end{bmatrix}
\]
TRIANGULAR MATRIX
A square matrix is called upper triangular if it has zeroes below the main diagonal. A square matrix is called lower triangular if it has zeroes above the main diagonal.
\[
\text{e.g.} \quad A = \begin{bmatrix}
5 & -1 & 9 \\
0& 3 & 8\\
0 & 0 & 1 \\
\end{bmatrix}
\] A is upper triangular
\[
\text{e.g.} \quad B= \begin{bmatrix}
3 & 0 & 0 \\
2& 5 & 0\\
1 & 0 & 2 \\
\end{bmatrix}
\] B is lower triangular
TRANSPOSE OF A MATRIX
The transpose of a matrix is \(A\) is obtained by interchanging the rows and the columns of A (i.e. 1st row becomes 1st column and vice versa; 2nd row becomes second column and vice versa; etc.) it is denoted as \(A^T\).
Example: If \(A = \begin{bmatrix}
5 & 1 & 7 \\
2& -1 & 3\\
\end{bmatrix}\)
Solution: We interchange the rows and columns.
\[
A^T =\begin{bmatrix}
5 & 2 \\
1 & -1 \\
7 & 3 \\
\end{bmatrix}
\]
SYMMETRIC MATRIX
A square matrix is said to be symmetric if it is equal to its transpose (i.e. ,you can interchange the rows with the columns and the matrix remains unchanged).This means \(A = A^T\). For this to be true you must have you must have, \(a_{ij} = a_{ji}\) for all \(i\) and \(j\).
\[
A = \begin{bmatrix}
4 & 5 & 7 \\
5& 1 & -2\\
7& -2 & -9\\
\end{bmatrix}
\]
IDENTITY MATRIX
A square matrix is called an identity matrix if it only has 1's along the main diagonal and 0's everywhere else. An \(n \times n\) matrix is denoted by \(I_n\). Multiplying a matrix by an identity matrix does not change the original matrix.
\[
I_3 = \begin{bmatrix}
1 & 0 & 0 \\
0& 1 & 0\\
0& 0 & 1\\
\end{bmatrix}
\]
ZERO MATRIX
A square matrix is called a zero matrix if all of its entries are 0. An \(n \times n \) matrix zero matrix is denoted by \(0_n\). Multiplying a matrix by a zero matrix results in a zero matrix of the appropriate size.
\[
0_3 = \begin{bmatrix}
0 & 0 & 0 \\
0& 0 & 0\\
0& 0 & 0\\
\end{bmatrix}
\]
TRACE OF A MATRIX
For a square matrix, we can find something called the "trace", denoted tr(A), which is just the sum of the entries along the main diagonal.
Example: If \(A =
\begin{bmatrix}
4 & 5 & 7 \\
5& 1 & -2\\
7& -2 & -9\\
\end{bmatrix}
\), find the trace, tr(A).
Solution: We add entries along the main diagonal.
\[
\begin{equation}
\begin{split}
tr(A) & = a_{11} + a_{22} + a_{33}\\
& = 4 + 1 + (-9) \\
& = -4 \\
\end{split}
\end{equation}
\]
