Matrix algebra is a means of making calculations upon arrays of numbers (or data).
Most data sets are matrix-type
Why use it?
Matrix algebra makes mathematical expression and computation easier.
It allows you to get rid of cumbersome notation, concentrate on the concepts involved and understand where your results come from.
Definitions
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Scalar multiplication of matrices
Transpose of a matrix
Special matrices
Multiplication of two or more matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
Matrices – definitions
A matrixis a set of real or complex numbers (called elements) arranged in rows and columns to form a rectangular array.
A matrix having m rows and n columns is called an m × n matrix.
For example:
$\left(\begin{array}{ccc}5& 7& 2\\ 6& 3& 8\end{array}\right)$
is a 2 × 3 matrix.
A matrix is a rectangular array of numbers or symbolic elements
In many applications, the rows of a matrix will represent individuals cases (people, items, plants, animals,...) and columns will represent attributes or characteristics
The dimension of a matrix is its number of rows and columns, often denoted as r x c (r rows by c columns)
Can be represented in full form or abbreviated form:
scalar a scalar is a number (denoted with regular type: 1 or 22).
Vector A matrix with one column (column vector) or one row (row vector).
Row vector
A row vector consists of a single row. For example:
$\left(\begin{array}{ccc}5& 7& 2\\ 6& 3& 8\end{array}\right)$
Column vector
A column matrix consists of a single column. For example:
$\left(\begin{array}{c}6\\ 3\\ 8\end{array}\right)$
Double suffix notation
Each element of a matrix has its own address denoted by double suffices, the first indicating the row and the second indicating the column. For example, the elements of 3 × 4 matrix can be written as:
Matrix Notation
Where there is no ambiguity a matrix can be represented by a single general element in brackets or by a capital letter in bold type.
Equal matrices
Two matrices are equal if corresponding elements throughout are equal
Addition and Subtraction
Where
Addition and subtraction of matrices
Two matrices are added (or subtracted) by adding (or subtracting) corresponding elements. For example:
Multiplication of matrices
Scalar multiplication
Multiplication of two or more matrices
Scalar multiplication
To multiply a matrix by a single number (a scalar), each individual element of the matrix is multiplied by that number. For example;
That is:
Properties of matrix addition and scalar multiplication
Then :-
$A+BA=ABA+AA$
Commutative law for addition
$A+(B+C)=(A+B)+C$
Associative law for addition
$\left(cd\right)A=c\left(dA\right)$
Associative law for scalar multiplication
$1A=A$
Unit element for scalar multiplication
$c(A+B)=cA+cB$
Distributive law 1 for scalar multiplication
$(c+d)A=cA+dA$
Distributive law 2 for scalar multiplication
Properties of zero matrices
Then
the additive identity for all m×n matrices
the additive inverse of A
Transpose of a matrix
If a new matrix is formed by interchanging rows and columns the new matrix is called the transpose of the original matrix. For example, if:
A square matrix is symmetric if For example: $\left(\begin{array}{ccc}1& 2& 2\\ 2& 8& 9\\ 5& 9& 4\end{array}\right)$
A square matrix is skew-symmetric
if
For example$\left(\begin{array}{ccc}0& 2& 5\\ -2& 0& 9\\ -5& -9& 0\end{array}\right)$
Diagonal Matrices.
A diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero
Identity Matrix (Unit Matrix)
An identity matrix is a diagonal matrix where the diagonal elements all equal one.$I=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
It is named I and it comes in different sizes.$I2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$I3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
Show that AC=BC
$A=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]B=\left[\begin{array}{cc}2& 4\\ 2& 3\end{array}\right]C=\left[\begin{array}{cc}1& -2\\ -1& 2\end{array}\right]$
C is noninvertible,
(i.e., row 1 and row 2
are not independent)
Solve$AC=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& -2\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}-2& 4\\ -1& 2\end{array}\right]$ $BC=\left[\begin{array}{cc}2& 4\\ 2& 3\end{array}\right]\left[\begin{array}{cc}1& -2\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}-2& 4\\ -1& 2\end{array}\right]$
So AC=BC
But A≠ B
Matrix Multiplication Properties
AB does not necessarily equal BA
(BA may even be an impossible operation)
For example
A × B = C(2 × 3) × (3 × 2) = (2 × 2)
B × A = D(3 × 2) × (2 × 3) = (3 × 3)
Matrix multiplication is Associative
Multiplication and transposition
A popular matrix
Matrix Multiplication Examples
Matrix Inverse
Matrix Inverse: Needed to perform the “division” of 2 square matrices
In scalar terms A/B is the same as A * 1/B
When we want to divide matrix A by matrix B we simply multiply by A by the inverse of B
An inverse matrix is defined as
A matrix is a rectangular array of different numbers. we can do different operations like addition, multiplication or subtraction.
Matrix size is the number of rows and columns of matrix. the matrix with (m rows) and (n columns) is called m × n. the m & n are calles its dimensions. for example if matrix (A) has 4 rows and 3 columns. we said A is 4 × 3 matrix
The sum A+B of two m-by-n matrices A and B is calculated entrywise:
(A + B)_{i,j} = A_{i,j} + B_{i,j}, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
To add two matrices, it is required that the number of rows in the first matrix be equal to the number of rows in
the second matrix and the number of columns in the first matrix equals the number of columns in the second matrix.
The product cA of a number c and a matrix A is computed by multiplying every entry of A by c:
(cA)_{i,j} = c · A_{i,j}.
The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:
(A^{T})_{i,j} = A_{j,i}.
The condition same as in matrix addition. it is required that the number of rows in the first matrix be equal to the number of rows in
the second matrix and the number of columns in the first matrix equals the number of columns in the second matrix.
To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.