- Definitions

- The sum of matrices of the same kind

- Scalar multiplication

- Sums in math

- Multiplication of a row matrix by a column matrix

- Multiplication of two matrices A.B

- Properties of multiplication of matrices

- Next steps

A matrix is an ordered set of numbers listed rectangular form.

Example. Let A denote the matrix

[2 5 7 8] [5 6 8 9] [3 9 0 1]

This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.

We denote the element on the second row and fourth column with a_{2,4}.

If a matrix A has n rows and n columns then we say it's a square matrix.

In a square matrix the elements
a_{i,i} , with i = 1,2,3,... , are called diagonal elements.

Remark. There is no difference between a 1 x 1 matrix and an ordinary number.

The diagonal matrix is completely defined by the diagonal elements.

Example.

[7 0 0] [0 5 0] [0 0 6] The matrix is denoted by diag(7 , 5 , 6)

[2 5 -1 5]

[2] [4] [3] [0]

Matrix A and B are of the same kind if and only if

A has as many rows as B and A has as many columns as B

[7 1 2] [4 0 3] [0 5 6] and [1 1 4] [3 4 6] [8 6 2]

The n x m matrix B is the transposed matrix of the m x n matrix A if and only if

The ith row of A = the ith column of B for (i = 1,2,3,..m)

So a_{i,j} = b_{j,i}

The transposed matrix of A is denoted T(A) or A^{T}T [7 1 ] [7 0 3] [0 5 ] = [1 5 4] [3 4 ]

When all the elements of a matrix A are 0, we call A a 0-matrix.

We write shortly 0 for a 0-matrix.

An identity matrix I is a diagonal matrix with all the diagonal elements = 1.

[1] [1 0] [0 1] [1 0 0] [0 1 0] [0 0 1] ...

A scalar matrix S is a diagonal matrix whose diagonal elements all contain the same scalar value.

a_{1,1} = a_{i,i} for (i = 1,2,3,..n)

[7 0 0] [0 7 0] [0 0 7]

If A' is the opposite of A then a

A square matrix is called symmetric if it is equal to its transpose.

Then a_{i,j} = a_{j,i} , for all i and j.

[7 1 5] [1 3 0] [5 0 7]

A square matrix is called skew-symmetric
if it is equal to the opposite of its transpose.

Then a_{i,j} = -a_{j,i} , for all i and j.

[ 0 1 -5] [-1 0 0] [ 5 0 0]

To add two matrices of the same kind, we simply add the corresponding elements.

Consider the set S of all n x m matrices (n and m fixed) and
A and B are in S.

From the properties of real numbers it's immediate that

- A + B is in S
- the addition of matrices is associative in S
- A + 0 = A = 0 + A
- with each A corresponds an opposite matrix -A
- A + B = B + A

To multiply a matrix with a real number, we multiply each element with this number.

Consider the set S of all n x m matrices (n and m fixed).
A and B are in S; r and s are real numbers.

It is not difficult to see that:

r(A+B) = rA+rB (r+s)A = rA+sA (rs)A = r(sA) (rA)^{T}= r. A^{T}

Remark. In this html document, for convenience, we'll write the word sum instead of the sigma sign.

This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).

To multiply the row by the column, you have to multiply all the corresponding elements,
then make the sum of the results.

Example.

[1] [2 1 3]. [2] = [19] [5]

So the number of columns of A has to be equal to the number of rows of B.

The product C = A.B then is a (l x n) matrix.

The element of the i-th row and the j-th column of the product is found by multiplying the ith row of A by the jth column of B.

cExamples._{i,j}= sum_{k}(a_{i,k}.b_{k,j})

[1 2][1 3] = [5 7] [2 1][2 2] [4 8] [1 3][1 2] = [7 5] [2 2][2 1] [6 6] [1 1][2 2] = [0 0] [1 1][-2 -2] [0 0] [ 1, 3, 2 ] [ 3, -1, 4 ] [ 1, 16, 5 ] [ 4, 5, 3 ] [ -2, 3, 1 ] = [ 8, 23, 18 ] [ 2, 2, 1 ] [ 2, 4, -1 ] [ 4, 8, 9 ]From these examples we see that the product is not commutative and that there are zero divisors. Zero divisors are matrices different from a zero matrix, such that the product is a zero matrix.

Application

A matrix A is called idempotent if and only if A^{2} = A.

Given:

[1 b c] A = [0 0 2] [0 0 1]Find the set of all 3 x 3 matrices of type A such that A is idempotent.

Solution:

We calculate A^{2}.

[1 b 2c+2b] [0 0 2 ] [0 0 1 ] AAll requested matrices are^{2}= A <=> 2c + 2b = c <=> c = -2b

[1 b -2b] [0 0 2 ] with b inR[0 0 1 ]

Proof:

We'll show that an element of A(B.C) is equal to the corresponding element of (A.B)C

First we calculate the element of the ith row and jth column of A(B.C)

Let D denote B.C, then dNow we calculate the element of the ith row and jth column of (A.B)C_{k,j}= sum_{p}b_{k,p}.c_{p,j}(1) Let E denote A.D then e_{i,j}= sum_{k}a_{i,k}.d_{k,j}(2) (1) in (2) gives e_{i,j}= sum_{k}a_{i,k}.(sum_{p}b_{k,p}.c_{p,j}) <=> e_{i,j}= sum_{k,p}a_{i,k}.b_{k,p}.c_{p,j}So the element of the ith row and jth column of A(B.C) is sum_{k,p}a_{i,k}.b_{k,p}.c_{p,j}(3)

Let D' denote A.B, then d_{i,p}' = sum_{k}a_{i,k}.b_{k,p}(4) Let E' denote D'C then e_{i,j}' = sum_{p}d_{i,p}'.c_{p,j}(5) (4) in (5) gives e_{i,j}' = sum_{p}(sum_{k}a_{i,k}.b_{k,p}).c_{p,j}<=> e_{i,j}' = sum_{k,p}a_{i,k}.b_{k,p}.c_{p,j}So the element of the ith row and jth column of (A.B)C is sum_{k,p}a_{i,k}.b_{k,p}.c_{p,j}(6) From (3) and (6) => A(B.C) = (A.B)C

If A is a square matrix then E = E'.

(A.B)This theorem can be proved in the same way as above.^{T}= B^{T}.A^{T}

Example :

If we transpose the product

[ 2 4 ] [x] [ 3 8 ] [y]we get

[x y ] [ 2 3 ] [ 4 8 ]

A.0 = 0 = 0.A

If D = diag(a,b,c) then D.D = diag( aThis property can be generalized for D = diag(a,b,c,d,e,...,l).^{2}, b^{2}, c^{2}) D.D.D = diag( a^{3}, b^{3}, c^{3}) .....

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