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ENGG2013 Unit 7Non-singular matrix
and Gauss-Jordan eliminationJan, 2011.
Outline
• Matrix arithmetic– Matrix addition, multiplication
• Non-singular matrix• Gauss-Jordan elimination
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The love function: a normal case
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Boy 1
Boy 2
Boy 3
Boy 4
Boy 5
Girl A
Girl B
Girl C
Girl D
Girl E
Function LDomain Range
Boy 1
Boy 2
Boy 3
Boy 4
Boy 5
Girl A
Girl B
Girl C
Girl D
Girl E
DomainRange
L(Boy 1) = Girl A, but L’(Girl A) = Boy 4.
Function L’
The love function: a utopian case
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Boy 1
Boy 2
Boy 3
Boy 4
Boy 5
Girl A
Girl B
Girl C
Girl D
Girl E
Function LDomain Range
Boy 1
Boy 2
Boy 3
Boy 4
Boy 5
Girl A
Girl B
Girl C
Girl D
Girl E
Function L’Domain Range
This function L’ is the inverse of L
The love function: no inverse
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Boy 1
Boy 2
Boy 3
Boy 4
Boy 5
Girl A
Girl B
Girl C
Girl D
Girl E
Function LDomain Range
Boy 1
Boy 2
Boy 3
Boy 4
Boy 5
Girl A
Girl B
Girl C
Girl D
Girl E
Domain Range
This function L has no inverse
This is not a function
Undo-able
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Multiplied by
Rotate 90 degrees clockwise
Multiplied by
Rotate 90 degrees counter-clockwise
A matrix which represents a reversibleprocess is called invertible or non-singular.
Objectives
• How to determine whether a matrix is invertible?
• If a matrix is invertible, how to find the corresponding inverse matrix?
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MATRIX ALGEBRA
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Matrix equality
• Two matrices are said to be equal if1. They have the same number of rows and the same number
of columns (i.e. same size).2. The corresponding entry are identical.
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Matrix addition and scalar multiplication
• We can add two matrices if they have the same size
• To multiply a matrix by a real number, we just multiply all entries in the matrix by that number.
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Matrix multiplication
• Given an mn matrix A and a pq matrix B, their product AB is defined if n=p.
• If n = p, we define their product, say C = AB, by computing the (i,j)-entry in C as the dot product of the i-th row of A and the j-th row of B.
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mnpq
m q
Examples
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is undefined.
is undefined.
Square matrix• A matrix with equal number of columns and rows is
called a square matrix.• For square matrices of the same size, we can freely
multiply them without worrying whether the product is well-defined or not.– Because multiplication is always well-defined in this case.
• The entries with the same column and row index are called the diagonal entries.– For example:
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Compatibility with function composition
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Multiplied by
Multiplied by
Multiplied by
Order does matter in multiplication
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Multiplied by
Rotate 90 degrees
Multiplied by
Reflection around x-axis
Multiplied by
Reflection around x-axis
Multiplied by
Rotate 90 degrees
Are they the same?
Non-commutativity
• For real numbers, we have 35 = 53.– Multiplication of real numbers is commutative.
• For matrices, in general AB BA.– Multiplication of matrices is non-commutative.– For example
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Associativity
• For real numbers, we have (34)5 = 3(45).– Multiplication of real numbers is associative.
• For any three matrices A, B, C, it is always true that (AB)C = A(BC), provided that the multiplications are well-defined.– Multiplication of matrices is associative.
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INVERTIBLE MATRIX
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Identity matrix• A square matrix whose diagonal entries are all one, and off-diagonal entries are all zero,
is called an identity matrix.
• We usually use capital letter I for identity matrix, or add a subscript and write In if we want to stress that the size is nn.
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Multiplication by identity matrix
• Identity matrix is like a do-nothing process.– There is no change after multiplication by the
identity matrix
• IA = A for any A.• BI = B for any B.
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Multiplied by
is trivial
Invertible matrix• Given an nn matrix A, if we can find a matrix A’, such that
then A is said to be invertible, or non-singular.• This matrix A’ is called an inverse of A.
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Multiplied by
A
Multiplied by
A’
Multiplied by
In
Example
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Multiplied by
Rotate 90 CW
Multiplied by
Rotate 90 CCW
implies is invertible.
Matrix inverse may not exist
• If matrix A induces a many-to-one mapping, then we cannot hope for any inverse.
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has no inverse
Naïve method for computing matrix inverse
• Consider
• Want to find A’ such that A A’= I• Solve for p, q, r, s in
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Uniqueness of matrix inverse
• Before we discuss how to compute matrix inverse, we first show there is at most one A’ such that A A’ = A’ A = I.
• Suppose on the contrary that there is another matrix A’’ such that A A’’ = A’’ A = I.
• We want to prove that A’ = A’’.
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Proof of uniqueness
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Defining property of A’’
Multiply by A’ from the left
I times anything is the same thing
Matrix multiplication is associative
Defining property of A’
I times anything is the same thing
Notation
• Since the matrix inverse (if exists) is unique, we use the symbol A-1 to represent the unique matrix which satisfies
• We say that A-1 is the inverse of A.
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A convenient fact
• To check that a matrix B is the inverse of A, it is sufficient to check either 1. BA = I, or2. AB = I.
• It can be proved that (1) implies (2), and (2) implies (1).– The details is left as exercise.
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GAUSS-JORDAN ELIMINATION
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Row operation using matrix
• Recall that there are three kind of elementary row operations1. Row exchange2. Multiply a row by a non-zero constant3. Replace a row by the sum of itself and a
constant multiple of another row.
• We can perform elementary row operation by matrix multiplication (from the left).
• All three kinds of operation are invertible.kshum ENGG2013 30
Row exchange
• Example: exchange row 2 and row 3
Multiply the same matrix from the left again, we get back the original matrix.
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Multiply a row by a constant
• Multiply the first row by -1.
Multiply the same matrix from the left again, we get back the original matrix.
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Row replacement
• Add the first row to the second row
Multiply by another matrix from the left to undo
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Elementary matrix (I)• Three types of elementary matrices
1. Exchange row i and row j
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Row i
Row j
Col
.
jCol
.
i
Elementary matrix (II)2. Multiply row i by m
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Row i
Col
.
i
Elementary matrix (III)3. Add s times row i to row j
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Row i
Row j
Col
.
jCol
.
i
Row reduction
• A series of row reductions is the same as multiplying from the left a series of elementary matrices.
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…
E1, E2, E3, … are elementary matrices.
If we can row reduce to identity
• Then A is non-singular, or invertible.
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(Matrixmultiplication isassociative)
Gauss-Jordan elimination
• It is convenient to append an identity matrix to the right
• We can interpret it asIf we can row reduce A to the identity by a series of
row operationsthen we can apply the same series of row operations to I and obtain the inverse of A.
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Algorithm
• Input: an nn matrix A.• Create an n 2n matrix M
– The left half is A– The right half is In
• Try to reduce the expanded matrix M such that the left half is equal to In.
• If succeed, the right half of M is the inverse of A.• If you cannot reduce the left half of M to , then A
is not invertible, a.k.a. singular.
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Example
• Find the inverse of
1. Create a 36 matrix 2. After some row reductions
we get
• Answer: kshum ENGG2013 41