Matrices |
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5) Find A2002 , if |
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A = |
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Solution: |
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1 1 1 1 1 2 |
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A2 = A A = |
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0 1 |
0 1 0 1 |
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1 1 1 2 1 3 |
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= A A |
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0 1 0 1 |
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1 3 |
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= A A |
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, and so on. |
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0 1 |
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A |
2002 |
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2002 |
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1.3. Types of Matrices
In a square matrix A =||ai, j ||, the elements ai,i , with i = 1, 2, 3, ... , are called the diagonal matrix elements. The set of the entries ai,i forms the leading (or principle) diagonal of the matrix.
A square matrix A =||ai, j || |
is called a |
diagonal matrix, if off-diagonal |
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elements are equal to zero or, symbolically, |
ai, j |
= 0 for all i ≠ j : |
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L |
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1,1 |
a2,2 |
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A = |
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M |
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L an,n |
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Identity matrices I are square matrices such that |
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I A = A |
and |
A I = A . |
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Compare these matrix equalities with the corresponding property of real numbers: 1 a = a and a 1 = a .
Theorem: Any identity matrix I is a diagonal matrix whose diagonal elements are equal to unity:
1 |
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L 0 |
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L 0 |
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I = |
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L M |
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L 1 |
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This theorem is proved in the following section.
12
Matrices
Examples:
1) It is not difficult to verify that
1 0 |
a b a b |
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a b |
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1 0 a b |
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c d |
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Therefore, 1 |
0 |
is the identity matrix of the second order. |
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2) Let A =||ai, j |
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be any 2 ×3 matrix. Then |
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a1,1 |
a1,2 |
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a1,1 |
a1,2 |
a1,3 |
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and |
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A matrix is called a zero-matrix (0-matrix), if it consists of only zero elements: ai, j = 0 for each {i, j}.
In a short form, a zero-matrix is written as 0:
0 |
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0 ≡ M |
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By the definition of a zero-matrix,
0 A = A 0 = 0
and
A +0 = A,
that is, a zero-matrix has just the same properties as the number zero.
However, if the product of two matrices is equal to zero, it does not mean that at least one of the matrices is a zero-matrix.
For instance, both matrices, |
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and |
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matrices, while their product is a zero-matrix:
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1 3 |
2 0 |
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Matrices
A square matrix has a triangular form, if all its elements above or below the leading diagonal are zeros:
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all |
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ai, j = 0 for |
i > j or for |
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Examples: |
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Upper-triangular matrix |
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Lower-triangular matrix |
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A = |
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b c |
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Given an m ×n matrix A =|| ai, j ||, the transpose of A is the n ×m matrix AT =|| a j,i || obtained from A by interchanging its rows and columns.
This means that the rows of the matrix A are the columns of the matrix AT ; and vise versa:
( AT )i, j = a j,i . |
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For instance, the transpose of A = |
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A square matrix A =||ai, j || is called a symmetric matrix, if A is equal to the |
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transpose of A: |
A = AT |
ai, j = a j,i . |
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The examples below illustrate the structure of symmetric matrices: |
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a |
c |
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= ST . |
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S = d |
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A square matrix A =||ai, j || is called a skew-symmetric matrix, if A is equal
to the opposite of its transpose:
ai, j = −a j,i .
The example below shows the structure of a skew-symmetric matrix:
0 |
−3 |
1 |
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A = |
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14 |
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Matrices
1.4. Kronecker Delta Symbol
The Kronecker delta symbol is defined by the formula
1, |
if |
i = j, |
δi, j = |
if |
i ≠ j. |
0, |
The delta symbol cancels summation over one of the indexes in such expressions as
∑aiδi, j , |
∑a jδi, j , |
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∑ai, jδi, j |
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∑ai, jδi, j , and so on. |
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For instance, the sum ∑aiδi, j |
may |
contain only one nonzero term |
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a jδ j, j = a j , while all the other terms are equal to zero, because of δi, j = 0 for any i ≠ j .
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n |
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If |
k ≤ j ≤ n , then ∑aiδi, j = a j . |
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i =k |
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If j < k |
or j > n , |
then |
∑aiδi, j = 0. |
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i =k |
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Likewise, if k ≤i ≤ n , then |
∑a jδi, j |
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j =k |
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Otherwise, if i < k or i > n , |
then ∑a jδi, j |
= 0 . |
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j =k |
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Examples: |
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100 |
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∑i2δi,3 |
=12 0 +22 0 |
+32 1+42 0 +K= 9 . |
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i =1 |
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∑2k δk ,10 |
= 210 =1024 , |
however ∑2k δk ,10 |
= 0 . |
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k =1 |
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Now we can easily prove the above-mentioned theorem of identity matrix:
1 |
0 |
L 0 |
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0 |
1 |
L 0 |
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I =||δi, j ||= |
M |
L |
O M |
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0 |
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L 1 |
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15 |
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