Matrix Concepts and Types

Study Notes

A matrix is a rectangular array of elements, which can include symbolic expressions or numbers.
Denoted mathematically as [ A], each entry is referred to as an element and is represented as aij, where i is the row number and j is the column number.

Row Vector: A matrix with one row, denoted as [ B] = [b1 b2 ... bn].
Column Vector: A matrix with one column, denoted as
```
[C] = 
[ c1 ]
[ c2 ]
[ ... ]
[ cm ]
```
Square Matrix: A matrix where the number of rows m is equal to the number of columns n. Diagonal elements are referred to as a11, a22, ..., ann.
Zero Matrix: A matrix where all elements are zero.
Diagonal Matrix: A square matrix where all off-diagonal elements are zero.
Identity Matrix: A diagonal matrix where all diagonal elements are 1.
Upper Triangular Matrix: A matrix where all elements below the main diagonal are zero.
Lower Triangular Matrix: A matrix where all elements above the main diagonal are zero.
Tri-diagonal Matrix: A matrix that only has non-zero elements on the main diagonal and the diagonals directly above and below it.
Dominant Matrix: A matrix where each diagonal element is greater than the sum of the absolute values of the other elements in its row.

The size of a matrix [A] is denoted by m×n, indicating m rows and n columns. For a matrix with 3 rows and 4 columns, the size is 3×4.

Two matrices are equal if they have the same dimensions and corresponding elements are equal.

A submatrix is formed by deleting some rows and/or columns from a matrix, without removing any rows or columns completely.

Illustration of tire sales data represented in matrix form showing sales by tire brands over four quarters.
Example matrices include a row vector [ B] = [25 20 3 2 0] and a column vector
```
[C] = 
[25]
[5]
[6]
```

Identifying sales in a matrix, such as finding the number of Copper tires sold in Quarter 4 by referencing the corresponding row and column.
Key features of specific matrix types like identifying upper triangular matrices through the positioning of zeros below the diagonal.