Download Matrix operations

Matrix operations
We define three operations in these sections: matrix-matrix addition, scalar-matrix multiplication, and matrixmatrix multiplication. It is important to note what the operators are operating on (sometimes it’s a scalar and a
matrix, other times it’s a matrix and a matrix), and what the result is.
Matrix - matrix addition
We define addition of two matrices in the following way: Consider a matrix A with elements indexed ai,j , and a
matrix B with elements bi,j . The matrix sum A + B is a matrix C is a matrix whose elements are computed by
ci,j = ai,j + bi,j .
• In English, that says add two matrices by adding the corresponding elements.
• In order for this definition to make sense, the matrices must be exactly the same size; if A is m × n, then B
must also be m × n for the sum A + B to be defined.
1 −1 2
5 −7 3
• Example: Let A =
. Let B =
. Find the sum A + B.
0 3 4
−2 6 1
Properties of matrix-matrix addition:
A, B, and C are all matrices. It is assumed that A, B and C all have the same size, so that addition can be performed.
• A + B = B + A (commutative property of addition)
• A + (B + C) = (A + B) + C (associative property of addition)
• A + 0 = 0 + A = A (additive identity property. The 0 is a 0 matrix)
• A + (−A) = (−A) + A = 0 (additive inverse property)
An additive identity is an object that when added to another object leaves the value of that object unchanged. For
example, the additive identity for real number addition (scalar addition) is 0, because x + 0 = x for any real number
x. Matrices have something analogous – the zero matrix serves as an additive identity: A + 0 = A. It is a matrix
whose elements are all zero, and whose dimensions are whatever you need them to be to make sense. For example,
1 2
0 0
1 2
+
=
3 4
0 0
3 4
An additive inverse is an object that when added to another object, returns the additive identity (giving the effect
of cancellation). Additive inverses for real numbers are negatives: x + (−x) = 0; and the same holds for matrices. We
define the negative of a matrix A, denoted B = −A, as a matrix where bi,j = −ai,j . For example,
1 −2
−1 2
0 0
+
=
−3 4
3 −4
0 0
Proving the properties (there’s a posted live example):
How would you prove that matrix-matrix addition is commutative?
This is what we’ll call a typical element proof. Matrix-matrix addition is defined in terms of what happens to a
generic element ci,j . If we can show that two matrices have exactly the same typical element, then they must be the
same matrix.
I like to do these by introducing a little extra notation. Assign each side of the equality its own matrix. Let
X = A+B
Y = B +A
the goal then is to show that X = Y by showing that xi,j = yi,j . Since the single elements xi,j and yi,j are real
numbers, we can apply the properties of real numbers to them.
Here we go ... (go watch the posted example)
Finally, the properties we are looking at are simply the group properties. The set of m × n matrices with the addition
operator
• is closed
• satisfies the associative axiom
• contains an identity element
• contains the additive inverse of each element
Furthermore, the addition operation is commutative. Therefore, for a given m, n, the set of m × n matrices with
addition forms an Abelian group, denoted < Mm,n , + >, or simply < Mm,n >.
Between the examples, the suggested problems, and the graded assignment, you will have proven that this is an
Abelian group by proving all the properties.