Business Mathematics (BK/IBA)
Quantitative Research Methods I (EBE)
Matrices
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Matrices
Contents
• Matrices
• Special matrices
• Operations with matrices
• Matrix multipication
• Matrix transposition
• Symmetric matrices
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Matrices
Matrices
• A matrix is a rectangular array of numbers or variables
• Notation
• we often use roman bold capital letters to refer to them
, , ⋯ ,
3
2
, , ⋯ ,
• e.g., = −2
,=
0
⋯
⋯
⋯
⋯
12.5 −12.7
, , ⋯ ,
• Terminology
•
•
•
•
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these matrices consist of 6 respectively elements
the order (or size is) 3 × 2 respectively × when = , the matrix is a square matrix
when ≠ , the matrix is rectangular
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Matrices
Matrices
• Notice the order of the indices
3
2
• In the matrix = −2
the element , refers
0
12.5 −12.7
to the cell at row 2 and column 1
• so to −2
• while , is in row 1 and column 2 and has value 2
• The order of is 3 × 2, not 2 × 3
• Convention :
• !,"#$% ! , × "#$%
• when no ambiguity you may skip comma: &' instead of &,'
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Matrices
Special matrices
• Zero matrix: ( =
0
0
⋯
0
0
0
⋯
0
⋯ 0
⋯ 0
⋯ ⋯
⋯ 0
• for a matrix of any order
• Identity matrix: ) =
• for a square matrix
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1 0 ⋯ 0
0 1 ⋯ 0
⋯ ⋯ ⋯ ⋯
0 0 ⋯ 1
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Matrices
Operations with matrices
• We can define some basic operations with matrices,
similar to the basic operations with vectors
•
•
•
•
•
addition ( + +, through + , &' = &' + ,&' )
multiplication (-, through - &' = - × &' )
negative matrix (−, through − &' = −&' )
subtraction ( − +, through − , &' = &' − ,&' )
equality ( = +, through &' = ,&' )
• But what about the inner product?
• not available for matrices
• instead: matrix multiplication
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Matrices
Matrix multiplication
• Let and + be two matrices, of order × . respectively
.×
• We define the matrix product + as
+
1
&'
= / &0 ,0' , 3 = 1, … , , 5 = 1, … , 02
• alternatively written as ⋅ +
• but do not write × +
• Notice well: the result of the multiplication of two matrices
is a matrix
• different for the inner product of two vectors
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Matrices
Matrix multiplication
• Illustration:
• +
• +
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∑
=
,
02 ,0 ,0, = , ,, + , ,,
8,8 = ∑02 8,0 ,0,8 = 8, ,,8 + 8, ,,8
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Matrices
Matrix multiplication
• Notice well the orders of the matrices:
• 1 + 1 = + • so #columns in should match #rows in +
• and #rows in + is #rows in • and #columns in + is #columns in +
• Consequences: given a matrix or order 3 × 3 and a
matrix + of order 3 × 2
• + exists and is of order 3 × 2
• + does not exist
• what about ? and ++? and + + ?
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Matrices
Matrix multiplication
• It follows that (with suitable , +, and 9)
• + + 9 = + + 9 (distributive property)
• + 9 = +9 = +9 (associative property)
• But not that
• + = + (commutative property)
1 2
2 −8
and + =
• example: take =
3 6
−1 4
• + =
0 0
−22 −44
, but + =
11
22
0 0
• By the way, notice that:
• + = (, while ≠ ( and + ≠ (
• while for numbers , = 0 ⇔ = 0or, = 0
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Matrices
Matrix multiplication
• Some properties (for suitable , +, and 9):
• ( = ( and ( = (
• ) = and ) = • + = 9 ⇏ + = 9
• example: =
• + = 9 =
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1 2
3 −4
1
4
,+=
, and 9 =
3 6
−2 3
−1 −1
−1 2
−3 6
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Matrices
Matrix multiplication
• What about powers of a matrix?
• Let us define = for any square matrix • why square?
• mind the differenve between “a square matrix” and “a
squared matrix”
• Likewise 8 = , etc.
• what is : is it or is it ?
• More in
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general = @ A
12
=1
= 2,3, …
Matrices
Matrix transposition
• Consider = × =
,
,
⋯
,
,
,
⋯
,
⋯ ,
⋯ ,
⋯
⋯
⋯ ,
• The transpose of , denoted by B is given by
, , ⋯ ,
, , ⋯ ,
B
= ⋯
⋯ ⋯
⋯
, , ⋯ ,
• In words, B has rows and Bcolumns so B is a ( × )-matrix
and row 3 of is column 3 of • “reflection in the diagonal”
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Matrices
Matrix transposition
• Some properties (for suitable , +, and 9):
• B B = • + + B = B + + B
• + B = + B B and +9 B = 9B + B B
• - B = -B
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Matrices
Symmetric matrices
• Definition
• The matrix is symmetric if and only if = B
• so if and only if &' = '& for all 3, 5
• note: only a square matrix can be symmetric
1 3
• Example: =
is symmetric
3 6
1 2 −1
• If =
the B is symmetric
3B 6 5
B
• note: is symmetric too but in general ≠ B (you can
see this for the example without even doing a calculation!)
• In general B is symmetric for an arbitrary matrix • and so is B (why?)
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Matrices