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Lecture 5
Andrei Antonenko
February 10, 2003
1 Matrices
Now we’ll start studying new algebraic object — matrices.
Definition 1.1. The matrix is a (rectangular) table of the elements of R. (Actually, we can consider matrices over fields other than R — in the future we will work with matrices over the field of complex numbers C.)
Now we’ll introduce some notation that we will use. We will denote matrices with capital letters, and the elements of the matrix with same small letter with 2 subscripts, the first of them denotes the row, and the second one denotes the column. Often we will speak about m × n-matrices, which means that it has m rows and n columns.
A = (aij ), A =
a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n ... ...... ...
am 1 am 2 · · · amn
The matrix is called square matrix if the number of its rows is equal to the number of its columns. For every square matrix we will define its main diagonal, or simply diagonal, as a diagonal from the top left corner to the bottom right corner, i.e. diagonal consists of the elements a 11 , a 22 ,... , ann. Another diagonal is called secondary. It is used very rarely. So, we introduced an object. But now we should introduce operations, otherwise the object is not interesting!
2 Matrix Operations
2.1 Addition
The first and the easiest matrix operation is matrix addition.
Definition 2.1. Let A and B are m × n-matrices. Then their sum C = A + B is an m × n- matrix such that cij = aij + bij , i.e. the elements of this matrix are sums of corresponding elements of the matrices A and B.
Example 2.2. (^) ( 1 2 3 3 0 − 1
Now let’s consider the properties of addition.
(A1) Commutativity. It is obvious that for any matrices A and B of the same size A + B = B + A.
(A2) Associativity. It is obvious that for any matrices A, B and C of the same size (A + B) + C = A + (B + C). Here we can mention that we can choose any order of matrices to perform the addition of 3 or more matrices. For example, we can prove, that (A + B) + C = (A + C) + B.
Proof.
(A + C) + B = (C + A) + B commutativity = C + (A + B) associativity = (A + B) + C commutativity
(A3) Existence of the zero. There exists a zero matrix
For any matrix A we have that A + 0 = A.
(A4) Existence of the additive inverse. For any matrix A there exists matrix −A such that A + (−A) = 0. The elements of this matrix (−a)ij = −aij.
Example 2.3. The additive inverse for the matrix
is
Example 2.6 (Cute example of matrix multiplication). Let α and β be real numbers, and let’s compute the following product ( cos α − sin α sin α cos α
cos β − sin β sin β cos β
It is equal to: ( cos α cos β − sin α sin β − cos α sin β − sin α cos β sin α cos β + cos α sin β − sin α sin β + cos α cos β
cos(α + β) − sin(α + β) sin(α + β) cos(α + β)
So,we can see that we get a matrix of the same type, but instead of α and β we have α + β.
Now let’s consider the properties of multiplication.
(M1) Commutativity. Unfortunately, commutativity does not hold for matrix multiplica- tion. Moreover, for some matrices A and B we can compute AB and cannot compute BA. E.g., if A is a 2 × 3-matrix, and B is a 3 × 3-matrix, then AB is defined, and BA is not. Now, we can give a counterexample even if both products are defined. ( 1 0 0 0
, but
Moreover, from this example we see that the product of two nonzero matrices can be a zero matrix.
(M2) Associativity. Associativity holds for matrix multiplication, i.e. for any three matrices such that all needed products(i.e., AB and BC) can be defined, we have that (AB)C = A(BC).
Proof. Let A = (aij ), B = (bij ), C = (cij ). Then (AB)ik =
l
ailblk, and
((AB)C)ij =
k
(AB)ikckj =
k
l
ailblkckj
In the same way (BC)lj =
k
blkclj , and so
(A(BC))ij =
l
ail(BC)lj =
l
ail
k
blkckj
Now we can change the order of the summation, and see that these expressions are equal.
Here, unlike in the case of addition, we cannot choose any order, since commutativity does not hold for multiplication. For example,
(AB)C 6 = (CA)B, etc.
The order of the matrices one multiplies should be always preserved.
(M3) Existence of the one. There exists an identity matrix
I =
This n × n-matrix has 1’s on its main diagonal. For any m × n-matrix A we have that AI = IA = A.
Proof. Can be done directly from the definition of the matrix multiplication. Simply can check that
AI =
a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n ... ...... ...
am 1 am 2 · · · amn
a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n ... ...... ...
am 1 am 2 · · · amn
By the same arguments, IA = A So, this matrix plays the same role for matrices as a number 1 for numbers. By multiplying by identity matrix, we do not change the given matrix.
Next lecture we are going to talk about the multiplicative inverse for matrices.
