Matrices

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Matrices

• MSU CSE 260

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Outline

• Introduction
• Matrix Arithmetic
• Sum, Product
• Transposes and Powers of Matrices
• Identity matrix, Transpose, Symmetric matrices
• Zero-one Matrices
• Join, Meet, Boolean product
• Exercise 2.6

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Introduction
Definition A matrix is a rectangular array of
numbers.
element in ith row, jth column
Also written as A?aij?
m rows
m?n matrix
n columns
When mn, A is called a square matrix.
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Matrix Equality

• Definition Let A and B be two matrices.
• AB if they have the same number of rows and
  columns, and every element at each position in A
  equals element at corresponding position in B.

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Matrix Addition, Subtraction
Let A ?aij? , B ?bij? be m?n matrices.
Then A B ?aij bij?, and A - B ?aij -
bij?
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Matrix Multiplication
Let A be a m?k matrix, and B be a k?n matrix,
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Matching Dimensions
To multiply two matrices, inner numbers must
match
Otherwise, not defined.
2?3 3?4
2?4 matrix
have to be equal
2?4
2?3
3?4
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Multiplicative Properties
Note that just because AB is defined, BA may not
be. Example If A is 3?4, B is 4?6, then AB3?6,
but BA is not defined (4?6 . 3?4). Even if both
AB and BA are defined, they may not have the same
size. Even if they do, matrices do not commute.
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Efficiency of Multiplication
3?4
2?3

• a11b12 a12b22 a13b32 c12
• Takes 3 multiplications, and 2 additions for each
  element.
• This has to be done 2?4 (8) times (since product
  matrix is 2?4). So 2?4?3 multiplications are
  needed.
• (m?k) (k?n) matrix product requires m.k.n
  multiplications.

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Best Order?
Let A be a 20?30 matrix, B 30?40, C 40?10. (AB)C
or A(BC)?
(20?30 30?40) 40?10
32000
20?30 (30?40 40?10)
18000
So, A(BC) is best in this case.
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Identity Matrix
The identity matrix has 1s down the diagonal,
e.g.
For a m?n matrix A, Im A A In
m?m m?n m?n n?n
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Inverse Matrix

• Let A and B be n?n matrices.
• If ABBAIn then B is called the inverse of A,
  denoted BA-1.
• Not all square matrices are invertible.

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Use of Inverse to Solve Equations
Please note that a-1j is NOT necessarily (aj)-1.
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Transposes of Matrices
Flip across diagonal
Transposes are used frequently in various
algorithms.
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Symmetric Matrix
A is called symmetric.
is symmetric. Note, for A to be symmetric, is has
to be square.
is trivially symmetric...
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Examples
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Power Matrix

• For a n?n square matrix A, the power matrix is
  defined as
• Ar A ? A ? ? A
• r times
• A0 is defined as In.

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Zero-one Matrices

• All entries are 0 or 1.
• Operations are ? and ?.
• Boolean product is defined using
• ? for multiplication, and
• ? for addition.

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Boolean Operations
Terminology is from Boolean Algebra.Think join
is put together, like union, and meet is
where they meet, or intersect.
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Boolean Product
(Should be a dot)
Since this is ord, you can stop when you
find a 1
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Boolean Product Properties

• In general, A ? B ? B ? A
• Example

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Boolean Power

• A Boolean power matrix can be defined in exactly
  the same way as a power matrix. For a n?n square
  matrix A, the power matrix is defined as
• Ar A ? A ? ? A
• r times
• A0 is defined as In.

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Exercise 2.6