Chapter 1. Section 6
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Section 1.6 – Algebraic Properties of Matrix Operations
Homework (page 69) problems 1-25
Recall the Properties of Real Numbers:
• Match the following
Commutative for multiplication
5 ⋅1 = 5
Associative for multiplication
3 (5 + 4) = 3(5) + 3(4)
(5 ⋅ 4) ⋅ 3 = 5 ⋅ (4 ⋅ 3)
Multiplicative identity
Distributive
5 + (–5) = 0
Identity for multiplication
5+4=4+5
5⋅ 4 = 4⋅5
Commutative for addition
Associative for addition
5 ( 1/5) = 1
Identity for addition
5+0=5
Additive identity
(5 + 4) + 3 = 5 + (4 + 3)
• Of the above, which do you think hold true for matrices?
Commutative for multiplication
Associative for multiplication
Multiplicative identity
Distributive
Identity for multiplication
Commutative for addition
Associative for addition
Identity for addition
Additive identity
Properties of Matrix Operations:
• Theorem 7: The following are true for matrices (recall that size of the matrices are important):
A+B=B+A
(A + B) + C = A + (B + C)
There exists a unique matrix O (called the zero matrix) such that A + O = A
There exists a unique matrix P such that A + P = O
• We have already seen that AB = BA is NOT true for matrices
• Theorem 8: The following are also true for matrices (recall that size of the matrices are important)
A(BC) = (AB)C
For scalars r and s, r(sA) = (rs)A = s(rA)
r(AB) = (rA)B = A(rB)
• Theorem 9: The following are also true for matrices (recall that size of the matrices are important)
(A + B)C = AC + BC
A(B + C) = AB + AC
(r + s)A = rA + sA
r(A + B) = rA + rB
• Page 69, Exercises 2 and 4
C. Bellomo, revised 28-Aug-06
Chapter 1. Section 6
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Transpose of a Matrix:
• For any element aij in a matrix A, the transpose, denoted AT = (bij), where bij = aji for all i,j
⎡1 4⎤
⎡1 2 3⎤
T
• For example, for the matrix A = ⎢
, A = ⎢2 5⎥
⎥
⎢
⎥
⎣ 4 5 6⎦
⎢⎣ 3 6 ⎥⎦
•
•
•
•
•
Theorem 10:
(A + B)T = AT + BT
(AC)T = CT AT
(AT)T = A
A matrix A is symmetric if A = AT
An nxn matrix is called a square matrix
In a matrix, the values aii are called the main diagonal
Page 69, Exercise 8
The Identity Matrix:
• The nxn identity matrix, denoted as In is the square matrix with 1’s on the main diagonal and 0’s
everywhere else.
⎡1 0 0 ⎤
• For example, I 3 = ⎢ 0 1 0 ⎥
⎢
⎥
⎢⎣ 0 0 1 ⎥⎦
•
The identity matrix is the multiplicative identity for matrix multiplication. This means that
A I n = A = In A
Scalar Products and Vector Norms:
• A vector is an nx1 matrix, usually denoted with x and y .
NOTE: the book uses bold notation for vectors.
• The scalar product for two vectors is simply matrix multiplication
⎡ y1 ⎤
T
x y = [ x1 x2 x3 ] ⎢⎢ y2 ⎥⎥ = x1 y1 + x2 y2 + x3 y3
⎢⎣ y3 ⎥⎦
•
In a similar fashion,
x T x = [ x1
x2
⎡ x1 ⎤
x3 ] ⎢⎢ x2 ⎥⎥ = x12 + x2 2 + x32
⎢⎣ x3 ⎥⎦
•
The norm is given by x = x T x
•
For two vectors, the Euclidean distance is given by x − y = ( x − y )T ( x − y )
•
Page 69, Exercises 14 and 20
C. Bellomo, revised 28-Aug-06