**Linear Algebra**

**1 Basic concepts of matrices**

**1.1 Definitions of Matrices**

A matrix is a rectangular array of numbers enclosed in brackets.

The numbers in the array are called the entries or elements of the matrix.

The size of a matrix is described in terms of the number of rows (horizontal lines) and columns (vertical lines) it contains.

A general matrix has a from

: the number of rows; : the number of columns.

Note. If , we call an square matrix. Its diagonal containing the entries is called the main diagonal of .

A matrix that is not square is called a rectangular matrix.

**Example 1.1.**

Some examples of matrices are

The size of the above matrices are (read 3 by 2), and .

A matrix with only one column is called a column matrix (or a column vector), for example, the matrix

A matrix with only one row is called a row matrix (or a row vector), e.g. the matrix

**1.2 Equality of matrices**

Two matrices and are equal, if and only if they have the same size and the corresponding entries are equal, i.e. and so on.

**Example 1.2.**

Consider the matrices

If , then . But for all other values of , the matrices and are not equal since not all of their corresponding entries are equal. There is no value of for which since and have different sizes.

**Exercise 1.2.**

Consider

and . Find .

**2 Basic matrix operations**

**2.1 Matrix addition and subtraction**

If and are matrices of the same size, then the sum is the matrix obtained by adding the entries of to the corresponding entries of . Similarly, the difference is the matrix obtained by subtracting the entries of from the corresponding entries of .

Note. Matrices of different sizes cannot be added or subtracted.

**Example 2.1.**

Consider the matrices

Then

The expressions , , and are undefined because the matrices are of different sizes.

The matrix addition or subtraction has the following properties

- (commutative law)
- (associated law)

Here is zero matrix of size , that is, a matrix with all entries zero.

**Exercise 2.1.**

Find and .

**2.2 Scalar Multiplication**

If is any matrix and is any scalar, then the product is the matrix obtained by multiplying each entry of the matrix by . The matrix is said to be a scalar multiple of .

**Example 2.2.**

Let

Then

The scalar multiplication has the following properties

- (associated law)
- (distributive law)

**2.3 Matrix multiplication**

If is an matrix and is an matrix, then the product exists if and only if . is an matrix with entries

That is, to get , multiply each entry in the th row of by the corresponding entry in the th column of and then add these products.

Note.

- The number of columns of must be the same as the number of rows of in order to form the product .
- If this condition is not satisfied, the product is undefined.
- A convenient way to determine whether a product of two matrices is defined is to write down the size of and, to the right of it, write down the size of .
- If, as shown in the figure below, the inside numbers are the same, then the product is defined.
- The outside numbers then give the size of the product.

**Example 2.3.**

Matrix multiplication

(a) Products of two matrices

Here is and is , so that comes out , whereas is undefined.

(b) Products of a matrix and a row matrix

whereas

is undefined.

(c) Products of a row and a column matrix

Note. The matrix multiplication has the following properties

- In general matrix multiplication is not commutative, i.e. .
- does not necessarily imply or or .
- does not necessarily imply (even when ).
- , is any scalar
- =
- =
- =

**Exercise 2.3.**

Consider

Observe (a), (b), (c) and (e).

**2.4 Matrix transposition**

The transpose of an matrix is the matrix that results from interchanging the rows and columns of . That is, the first column of is the first row of , the second column of is the second row of , and so forth.

**Example 2.4.**

The following are some examples of matrices and their transposes.

The transposition has the following properties

**Exercise 2.4.**

Consider

Observe (d).

**2.5 Matrix trace**

If is a square matrix, then the trace of , denoted by , is defined to be the sum of the entries on the main diagonal of . The trace of is undefined if is not a square matrix.

**Example 2.5.**

**Exercise 2.5.**

Since , find .

**3 Special Matrices**

The following are certain kind of matrices that are frequently used.

**3.1 Symmetric matrices**

Symmetric matrices are square matrices whose transpose equals the matrix, i.e.

By inspection, the mirror images of the entries of symmetric matrices across the main diagonal must be equal. The product of any matrix and its transpose, i.e. or will always give symmetric matrices.

**Example 3.1.**

Symmetric matrices

Let be the matrix

Observe that and are symmetric matrices.

**3.2 Skew-symmetric matrices**

Skew-symmetric matrices are square matrices whose transpose equals the negative of the matrix, i.e.

The subtraction of a square matrix and its transpose, i.e. will always give skew-symmetric matrices.

**Example 3.2.**

Skew-symmetric matrices

Let be the matrix

Observe that is a skew symmetric matrix.

**3.3 Upper triangular matrices**

Upper triangular matrices are square matrices that can have nonzero entries only on and above the main diagonal, whereas any entry below the diagonal must be zero.

**Example 3.3.**

Upper triangular matrices

**3.4 Lower triangular matrices**

Lower triangular matrices are square matrices that can have nonzero entries only on and below the main diagonal, whereas any entry above the diagonal must be zero.

**Example 3.4.**

Lower triangular matrices

**3.5 Diagonal matrices**

Diagonal matrices are square matrices that can have nonzero entries only on the main diagonal . Any entry above or below the main diagonal must be zero.

Special kind of diagonal matrices:

- Scalar matrix S is a diagonal matrix with all the diagonal entries equal to . Multiplication of any square matrix of the same size by S has the same effect as the multiplication by a scalar.

- Identity or unit matrix I is a diagonal matrix with all the diagonal entries equal to 1.

**Example 3.5.**

Diagonal matrix , scalar matrix , unit matrix

**4 Basic concepts of vectors**

**4.1 Definitions of vectors**

A vector is a matrix that has only one row or one column. If the matrix has one row then we call the matrix a row vector. It is of the form

If the matrix has one column then we call the matrix a column vector. It is of the form

The matrix transposition converts a row vector to a column vector and vice versa.

**Example 4.1.**

and are row vectors. and are column vectors.

**4.2 Inner product of vectors**

If is a row vector and a column vector, both with components, then matrix multiplication (row times column) gives a matrix or a scalar. This product is called the inner product or dot product of and and is denoted by . Thus,

**Example 4.2.**

Let and then

**Exercise 4.2.**

Let and then find .

**4.3 Matrix multiplication in terms of row and column vectors**

Matrix multiplication is a multiplication of rows of into columns of . Thus, we can write the product in terms of inner products of row and column vectors. This can be explained as follows.

An matrix can be represented by a column matrix which elements are row vectors.

An matrix can be represented by a row matrix which elements are column vectors.

Suppose that . Then we have

and

**Example 4.3.**

The row vectors , , of the matrix

The column vector , of the matrix

Hence from (2) we obtain the product

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