Download Identity Matrices Powers and Transposes of Matrices

2/21/2017
Identity Matrices
Powers and Transposes of Matrices
The identity matrix of order n is the n×n matrix
In = [δij], where δij = 1 if i = j and δij = 0 if i ≠ j:
The power function can be defined for square
matrices. If A is an n×n matrix, we have:
A0 = In,
Ar = AAA…A (r times the letter A)
The transpose of an m×n matrix A = [aij], denoted by
At, is the n×m matrix obtained by interchanging the
rows and columns of A.
Multiplying an m×n matrix A by an identity matrix of
appropriate size does not change this matrix:
AIn = ImA = A
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In other words, if At = [bij], then bij = aji for
i = 1, 2, …, n and j = 1, 2, …, m.
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A matrix with entries that are either 0 or 1 is called a
zero-one matrix. Zero-one matrices are often used
like a “table” to represent discrete structures.
Example:
We can define Boolean operations on the entries in
zero-one matrices:
A square matrix A is called symmetric if A = At.
Thus A = [aij] is symmetric if aij = aji for all
i = 1, 2, …, n and j = 1, 2, …, n.
A is symmetric, B is not.
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a
b
a∧b
a
b
a∨b
0
0
0
0
0
0
0
1
0
0
1
1
1
0
0
1
0
1
1
1
1
1
1
1
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Zero-One Matrices
Then the join of A and B is the zero-one matrix with
(i, j)th entry aij ∨ bij. The join of A and B is denoted by
A ∨ B.
The meet of A and B is the zero-one matrix with (i,
j)th entry aij ∧ bij. The meet of A and B is denoted by
A ∧ B.
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Week 5: Mathematical Reasoning
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Zero-One Matrices
Let A = [aij] and B = [bij] be m×n zero-one matrices.
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Zero-One Matrices
Powers and Transposes of Matrices
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Example:
Join:
Meet:
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Zero-One Matrices
Zero-One Matrices
Let A = [aij] be an m×k zero-one matrix and
B = [bij] be a k×n zero-one matrix.
Then the Boolean product of A and B, denoted by
AοB, is the m×n matrix with (i, j)th entry [cij], where
Example:
cij = (ai1 ∧ b1j) ∨ (ai2 ∧ b2i) ∨ … ∨ (aik ∧ bkj).
Note that the actual Boolean product symbol has a
dot in its center.
Basically, Boolean multiplication works like the
multiplication of matrices, but with computing ∧
instead of the product and ∨ instead of the sum.
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February 21, 2017
Zero-One Matrices
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Let’s proceed to…
Let A be a square zero-one matrix and r be a positive
integer.
The r-th Boolean power of A is the Boolean product
of r factors of A. The r-th Boolean power of A is
denoted by A[r].
A[0] = In,
A[r] = AοAο…οA
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Mathematical Reasoning
(r times the letter A)
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Terminology
Mathematical Reasoning
We need mathematical reasoning to
An axiom is a basic assumption about mathematical
structures that needs no proof.
• determine whether a mathematical argument is
correct or incorrect and
• construct mathematical arguments.
We can use a proof to demonstrate that a particular
statement is true. A proof consists of a sequence of
statements that form an argument.
Mathematical reasoning is not only important for
conducting proofs and program verification, but
also for artificial intelligence systems (drawing
inferences).
The steps that connect the statements in such a
sequence are the rules of inference.
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Cases of incorrect reasoning are called fallacies.
A theorem is a statement that can be shown to be
true.
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Terminology
Rules of Inference
A lemma is a simple theorem used as an
intermediate result in the proof of another theorem.
A corollary is a proposition that follows directly from
a theorem that has been proved.
A conjecture is a statement whose truth value is
unknown. Once it is proven, it becomes a theorem.
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Rules of inference provide the justification of the
steps used in a proof.
One important rule is called modus ponens or the
law of detachment. It is based on the tautology
(p∧(p→q)) → q. We write it in the following way:
p
p→q
____
∴q
The two hypotheses p and p → q are
written in a column, and the conclusion
below a bar, where ∴ means “therefore”.
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Rules of Inference
The rule states that if p1 and p2 and … and
pn are all true, then q is true as well.
These rules of inference can be used in
any mathematical argument and do not
require any proof.
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Rules of Inference
The general form of a rule of inference is:
p1
p2
.
.
.
pn
____
∴q
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p
_____
∴ p∨q
Addition
¬q
p→q
_____
∴ ¬p
p∧q
_____
∴p
Simplification
p→q
q→r
_____
∴ p→r
Hypothetical
syllogism
p
q
_____
∴ p∧q
Conjunction
p∨q
¬p
_____
∴q
Disjunctive
syllogism
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Applied Discrete Mathematics
Week 5: Mathematical Reasoning
Modus tollens
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