Theorem 4.1 The Basic Rules of Matrix Algebra)
4.3 Different Types of Square Matrices
You have already encountered many types of square matrices. Some of them have special uses or properties, so it is helpful to refer to them by special names.
In this lesson, you will learn how to
• recognize the different types of square matrices
• decompose any square matrix into its symmetric and skew-symmetric parts
A square matrix is ann×nmatrix: a matrix with the same number of rows and columns. Here is a generic square matrix:
A=
⎛
⎜⎜
⎜⎜
⎜⎜
⎝
A11 A12 A13 ã ã ã A1n
A21 A22 A23 ã ã ã A2n
A31 A32 A33 ã ã ã A3n
... ... ... . .. ... An1 An2 An3 ã ã ã Ann
⎞
⎟⎟
⎟⎟
⎟⎟
⎠
Square matrices are common enough that they warrant further classifica- tion. Here are some special kinds of square matrices. The examples are all 3×3 or 4×4, but these special types of square matrices can occur for any n×n.
Special type
Example Description
Identity
matrix I=
⎛
⎝1 0 0 0 1 0 0 0 1
⎞
⎠ Iij=
1 ifi=j 0 otherwise
Diagonal matrix
⎛
⎝3 0 0 0 5 0 0 0 π
⎞
⎠ Aij = 0 wheni=j
Scalar matrix
⎛
⎝3 0 0 0 3 0 0 0 3
⎞
⎠ Aij = 0 wheni=j, and c=A11=A22=A33=ã ã ã
Note:A=cI
4.3 Different Types of Square Matrices
Special type
Example Description
Upper triangular
matrix
⎛
⎝1 2 3 0 4 5 0 0 6
⎞
⎠ Aij = 0 wheni > j
Lower triangular
matrix
⎛
⎜⎜
⎝
1 0 0 0 2 3 0 0 4 5 6 0 7 8 9 10
⎞
⎟⎟
⎠ Aij = 0 wheni < j
Symmetric matrix
⎛
⎜⎜
⎝
1 2 3 4 2 5 6 7 3 6 8 9 4 7 9 10
⎞
⎟⎟
⎠
Aij=Aji
or Ak∗=Ak∗
or A=A
Skew- symmetric
matrix
⎛
⎜⎜
⎝
0 1 2 3
−1 0 4 5
−2 −4 0 6
−3 −5 −6 0
⎞
⎟⎟
⎠
Aij =−Aji
or Ak∗=−Ak∗
or A =−A
For You to Do 1. LetA =
⎛
⎝1 2 3
4 6 9
5 −2 0
⎞
⎠. Calculate A+A and A−A. What kind of matrices are these?
Your work on problem 1 suggests the following lemma.
Lemma 4.2
IfA is a square matrix, then
• A+A is symmetric, and
• A−A is skew-symmetric
There are multiple possible proofs of the lemma, including ones that track A through its entries, or its rows and columns. One simple proof uses only the definitions of symmetric and skew-symmetric, along with some properties of the transpose operator you explored in For Discussion problem 5 from Lesson 4.2.
161
Proof. A+A is symmetric if and only if its transpose equals itself, and it does:
(A+A)=A+ (A)=A+A=A+A
The proof thatA−A is skew-symmetric is almost identical. A matrix is skew-symmetric if and only if its transpose is also its negative.
(A−A)=A−(A)=A−A=−A+A=−(A−A) Symmetry and skew-symmetry are preserved under scalar multiplica- tion, as the following lemma states.
Lemma 4.3
IfAis symmetric, thencAis also symmetric. IfAis skew-symmetric, then cAis also skew-symmetric.
You will be asked to prove this lemma in Exercise 2.
Look back at Exercise 6c from Lesson 4.2. There, you were asked to calculate
1
2(A+A) +12(A−A)
This expression turned out to equal the original matrix A. But this expression is also the sum of a symmetric matrix and a skew-symmetric matrix, according to the lemmas above. The algebra is general, leading to the following theorem.
Theorem 4.4
Every square matrix is the sum of a symmetric matrix and a skew- symmetric matrix.
Proof. If the square matrix isA, the symmetric matrix is 12(A+A), and the skew-symmetric matrix is 12(A−A).
1
2(A+A)
symmetric
+ 12(A−A)
skew-symmetric
=12A+12A+12A−12A
=1
2A+12A +1
2A−12A
=A+O
=A For Discussion
2. a. Let A=
⎛
⎝ 3 4 2
−2 7 0 2 −1 5
⎞
⎠. Write A as the sum of a symmetric and a skew- symmetric matrix.
b. Explain whyB=
3 4 2
−2 7 0
cannot be written as the sum of a symmetric and a skew-symmetric matrix.
4.3 Different Types of Square Matrices
Exercises
1. Prove Lemma 4.2.
a. Use the entry-by-entry definitions of symmetric and skew- symmetric.
b. Use the row-by-row definitions of symmetric and skew-symmetric.
2. Prove the two parts of Lemma 4.3.
a. Use any definition of symmetric to show that ifAis symmetric andc is a scalar, thencAis symmetric.
b. Use any definition of skew-symmetric to show that ifAis skew- symmetric andc is a scalar, thencAis skew-symmetric.
3. If A = a b
c d
, write A as the sum of a symmetric matrix and a skew-symmetric matrix.
4. Theorem 4.4 gives a method to write a square matrixAas the sum of a symmetric matrix and a skew-symmetric matrix. Prove that this pair of matrices is unique, or find some other pairs of matrices that can also be used.
5. For each given matrixA, find an expression in terms ofiandjthat defines the entriesAij of the matrix. All matrices can be defined as
A=
⎛
⎝A11 A12 A13 A14
A21 A22 A23 A24
A31 A32 A33 A34
⎞
⎠
a. A=
⎛
⎝0 −1 −2 −3
1 0 −1 −2
2 1 0 −1
⎞
⎠ b. A=
⎛
⎝4 6 8 10 6 8 10 12 8 10 12 14
⎞
⎠
c. A=
⎛
⎝2 2 2 2 4 4 4 4 6 6 6 6
⎞
⎠ d. A=
⎛
⎝3 5 7 9 4 6 8 10 5 7 9 11
⎞
⎠
e.A=
⎛
⎝2 3 4 5
5 6 7 8
10 11 12 13
⎞
⎠
6. Classify each given matrix as scalar, diagonal, upper or lower triangular, symmetric, skew-symmetric, or none of the above.
a. Ais 3×3 andAij =i2+j2 b. Ais 4×4 andAij =i2−j2 c. Ais 3×3 andApq=p2+pq+q2 d. Ais 2×2 andApq=p−2q e. Ais 2×2 andArs= 2s−r f.A= rref
⎛
⎝1 2 3 4 5 6 7 8 9
⎞
⎠ g.A= rref
⎛
⎝1 2 3 4 5 6 7 8 0
⎞
⎠
163
h. A is 3×3 andAij =
i+j ifi > j 0 ifi≤j i. A is 4×4 andAij =
i2 ifi=j 0 ifi=j 7. Writeain terms ofx.
a= 2m−3n+ 5p m= 4x
n= 5x p=−x 8. Writeaandbin terms of x.
a= 2m−3n+ 5p b=−2m+ 6n+ 10p m= 4x
n= 5x p=−x 9. Writeain terms ofxandy.
a= 2m−3n+ 5p m= 4x−2y
n= 5x+ 3y p=−x+ 5y 10. Writeaandbin terms of x, y, z, andw.
a= 2m−3n+ 5p b=−2m+ 6n+ 10p m= 4x−2y+ 7z+w
n= 5x+ 3y−3z+w p=−x+ 5y+ 2z+w
4.3 Different Types of Square Matrices
11. As in Exercise 7 from Lesson 4.1, this 5×5 matrix gives the number of one-way, nonstop flights between four different cities served by an airline.
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎝
PDX LAS DEN CLT PHL
PDX 0 0 5 0 1
LAS 0 0 0 5 5
DEN 5 0 0 4 3
CLT 0 5 4 0 10
PHL 1 5 3 8 0
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎠
Philadelphia PHL
CLT Charlotte Denver DEN
LAS
1
1 3
3 5 5
5
5 5
5
4 4
4
10 8 Las Vegas
PDX Portland
a. How many different one-stop flights are there from Portland to Las Vegas?
←−There are 10 daily nonstop flights from Charlotte to Philadelphia.
b. How many different one-stop flights are there from Portland to Denver?
c. How many different one-stop flights are there from Portland to Charlotte?
d. How many different one-stop flights are there from Portland to Philadelphia?
e. How many different one-stop flights are there from Philadel- phia to Denver?
f. How many different two-stop flights are there from Philadel-
phia to Denver? ←−
Yes, one of those stops can be Denver or Philadelphia.
165