1 Introduction to Matrices

In this book, we shall mostly be concerned with complex numbers. The horizontal arrays of a matrix are called its rows and the vertical arrays are called its columns. Let A be a matrix having m rows and n columns. Then, A is said to have order m × n or is called a matrix of size m × n and can be represented in either of the following forms:

For example, if A = [ ]
1 3+ i 7

4 5 6 - 5i

then A[1,:] = [13 + i7], A[:,3] = [ ]
7

6- 5i

and a₂₂ = 5. In general, in row vector commas are inserted to differentiate between entries. Thus, A[1,:] = [1,3 + i,7]. A matrix having only one column is called a column vector and a matrix with only one row is called a row vector. All our vectors will be column vectors and will be represented by bold letters. Thus, A[1,:] is a row vector and A[:,3] is a column vector.

PICT PICT DRAFT Example 1.1.2. The system of linear equations 2x+3y = 5 and 3x+2y = 6 can be identified with the matrix A = [2 3 |5]
|
3 2 |6 . Note that x and y are variables with the understanding that x is associated with A[:,1] and y is associated with A[:,2].

Definition 1.1.3. Two matrices A = [a_ij],B = [b_ij] ∈ M_m,n(ℂ) are said to be equal if a_ij = b_ij, for each i = 1,2,…,m and j = 1,2,…,n.

In other words, two matrices are said to be equal if they have the same order and their corresponding entries are equal.

1.1.1 Special Matrices

Definition 1.1.4.

1.

A matrix in which each entry is zero is called a zero-matrix, denoted 0. For example,

[ ] [ ] 02×2 = 0 0 and 02×3 = 0 0 0 . 0 0 0 0 0

2.

A matrix that has the same number of rows as the number of columns, is called a square matrix. A square matrix is said to have order n if it’s order is n × n and is denoted either by writing A ∈ M_n(ℝ) or A ∈ M_n(ℂ), depending on whether the entries are real or complex numbers, respectively. PICT

DRAFT

3.

Let A = [a_ij] ∈ M_m,n(ℂ).

(a): Then, the entries a₁₁,a₂₂,…,a_nn are called the diagonal entries and they constitute the principal diagonal of A.
(b): Then, A is said to be a diagonal matrix if a_ij = 0 for i≠j, denoted diag(a₁₁,…,a_nn). For example, the zero matrix 0_n and are two diagonal matrices.
(c): If A = diag(a₁₁,…,a_nn) and a_ii = d for all i = 1,…,n then the diagonal matrix A is called a scalar matrix.
(d): Then, A = diag(1,…,1) is called the identity matrix, denoted I_n, or in short I. For example, I₂ = and I₃ = .

4.

For 1 ≤ i ≤ n, define e_i = I_n[:,i], a matrix of order n × 1. Then, the set {e₁,…,e_n}, where e_i ∈ M_n,1(ℂ), for 1 ≤ i ≤ n, is called the standard basis of ℂⁿ. Note that even though the order of the column vectors e_i’s depend on n, we don’t mention it as the size is understood from the context. For example, if e₁ ∈ ℂ² then, e₁^T = [1,0]. If e₁ ∈ ℂ³ then, e₁^T = [1,0,0] and so on.

5.

Let A = [a_ij] be a square matrix.

(a): Then, A is said to be an upper triangular matrix if a_ij = 0 for i > j.
(b): Then, A is said to be a lower triangular matrix if a_ij = 0 for i < j.
(c): Then, A is said to be triangular if it is an upper or a lower triangular matrix.
For example, is upper triangular, is lower triangular and the matrices 0,I are upper as well as lower triangular matrices.

6.

An m × n matrix A = [a_ij] is said to have an upper triangular form if a_ij = 0 for all i > j. For example, the matrices ⌊ ⌋
a11 a12 ⋅⋅⋅ a1n
|| 0 a22 ⋅⋅⋅ a2n||
|| . . . .||
⌈ .. .. .. ..⌉
0 0 ⋅⋅⋅ a
nn

and

have upper PICT

DRAFT triangular forms.

1.2 Operations on Matrices

Definition 1.2.1. Let A = [a_ij] ∈ M_m,n(ℂ).

1.: Then, the transpose of A, denoted A^T = [b_ij] ∈ M_n,m(ℂ) and b_ij = a_ji, for all i,j.
2.: Then, the conjugate transpose of A, denoted A^* = [c_ij] ∈ M_n,m(ℂ) and c_ij = a_ji, for all i,j, where for a ∈ ℂ, a denotes the complex-conjugate of a.

Thus, if x is a column vector then x^T and x^* are row vectors and vice-versa. For example, if A = [ ]
1 4 5
0 1 2

then A^* = A^T = ⌊ ⌋
1 0
| 4 1|
⌈ ⌉
5 2

, whereas if A = [ ]
1 4 + i
0 1 - i

then A^T = [ ]
1 0
4+ i 1- i

and A^* =

. Note that A^*≠A^T.

Theorem 1.2.2. For any matrix A, (A^*)^* = A. Thus, (A^T)^T = A.

Proof. Let A = [a_ij],A^* = [b_ij] and (A^*)^* = [c_ij]. Clearly, the order of A and (A^*)^* is the same. Also, by definition c_ij = b_ji = a_ij = a_ij for all i,j and hence the result follows. _

Definition 1.2.3. Let A = [a_ij],B = [b_ij] ∈ M_m,n(ℂ). Then, the sum of A and B, denoted A + B, is defined to be the matrix C = [c_ij] ∈ M_m,n(ℂ) with c_ij = a_ij + b_ij. PICT PICT DRAFT

Definition 1.2.4. Let A = [a_ij] ∈ M_m,n(ℂ). Then, the product of k ∈ ℂ with A, denoted kA, is defined as kA = [ka_ij] = [a_ijk] = Ak.

Theorem 1.2.5. Let A,B,C ∈ M_m,n(ℂ) and let k,ℓ ∈ ℂ. Then,

1.: A + B = B + A (commutativity).
2.: (A + B) + C = A + (B + C) (associativity).
3.: k(ℓA) = (kℓ)A.
4.: (k + ℓ)A = kA + ℓA.

PICT PICT DRAFT Definition 1.2.6. Let A ∈ M_m,n(ℂ).

1.: Then, the matrix 0_m×n is called the additive identity as A + 0 = 0 + A = A.
2.: Then, there exists a matrix B with A + B = 0. This matrix B is called the additive inverse of A, and is denoted by -A = (-1)A.

Exercise 1.2.7.

1.

Find a few 3 × 3 nonzero, non-identity matrices A with real entries satisfying

(a): A^T = A.
(b): A^T = -A.

2.

Find a few 3 × 3 nonzero, non-identity matrices A with complex entries satisfying

(a): A^* = A.
(b): A^* = -A.

3.

Suppose A = [a_ij] and B are matrices such that A + B = 0. Then, show that B = (-1)A = [-a_ij].

4.

Suppose A and B are matrices such that A + B = A. Then, show that B = 0. _________

5.

Let A =

and B =

. Compute A + B^* and B + A^*.

6.

Write the 3 × 3 matrices A = [a_ij] satisfying PICT

DRAFT

(a): a_ij = 1 if i≠j and 2 otherwise.
(b): a_ij = 1 if |i - j|≤ 1 and 0 otherwise.
(c): a_ij = i + j.
(d): a_ij = 2^i+j.

1.2.1 Multiplication of Matrices

Definition 1.2.8. Let A = [a_ij] ∈ M_m,n(ℂ) and B = [b_ij] ∈ M_n,r(ℂ). Then, the product of A and B, denoted AB, is a matrix C = [c_ij] ∈ M_m,r(ℂ) with

∑n cij = aikbkj = ai1b1j + ai2b2j + ⋅⋅⋅ + ainbnj,1 ≤ i ≤ m, 1 ≤ j ≤ r. k=1

Thus, AB is defined if and only if number of columns of A = number of rows of B.

(AB)[:,2] = βA[:,1] + yA[:,2] + vA[:,3], ⋅⋅⋅

,(AB)[:,4] = δA[:,1] + tA[:,2] + sA[:,3].

Remark 1.2.9. Observe the following: PICT PICT DRAFT

1.

In this example, while AB is defined, the product BA is not defined. However, for square matrices A and B of the same order, both the product AB and BA are defined.

2.

The product AB corresponds to operating (adding or subtracting multiples of different rows) on the rows of the matrix B(see Equation (1.2.2)). This is row method for calculating the matrix product.

3.

The product AB also corresponds to operating (adding or subtracting multiples of different columns) on the columns of the matrix A (see Equation (1.2.3)). This is column method for calculating the matrix product.

4.

Let A and B be two matrices such that the product AB is defined. Then, verify that

(a): Then, verify that (AB)[i,:] = A[i,:]B. That is, the i-th row of AB is obtained by multiplying the i-th row of A with B.
(b): Then, verify that (AB)[:,j] = AB[:,j]. That is, the j-th column of AB is obtained by multiplying A with the j-th column of B.

Hence,

⌊ ⌋ A [1,:]B ||A [2,:]B|| AB = || . || = [AB [:,1],AB [:,2], ...,AB [:,p]]. ⌈ .. ⌉ A [n, :]B

(1.2.4)

DRAFT

Example 1.2.10. Let A = ⌊ ⌋
|1 2 0|
⌈1 0 1⌉
0 - 1 1 and B = ⌊ ⌋
| 1 0 - 1|
⌈ 0 0 1 ⌉
0 - 1 1 . Use the row/column method of matrix multiplication to

1.: find the second row of the matrix AB.
Solution: By Remark 1.2.9.4, (AB)[2,:] = A[2,:]B and hence $(AB )[2,:] = 1 ⋅[1,0,- 1]+ 0⋅ [0,0,1]+ 1 ⋅[0,- 1,1] = [1,- 1,0].$
2.: find the third column of the matrix AB.
Solution: Again, by Remark 1.2.9.4, (AB)[:,3] = AB[:,3] and hence $⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ 1 2 0 1 (AB )[:,3] = - 1⋅ |⌈1|⌉ + 1⋅|⌈ 0 |⌉+ 1 ⋅|⌈1|⌉ = |⌈0|⌉ . 0 - 1 1 0$

PICT PICT DRAFT Exercise 1.2.11.

1.

For 1 ≤ i ≤ n, recall the basis elements e_i ∈ M_n,1(ℂ) (see Definition 4). If A ∈ M_n(ℂ) then

(a): Ae₁ = A[:,1],…,Ae_n = A[:,n].
(b): e₁^TA = A[1,:],…,e_n^TA = A[n,:].

2.

Let A ∈ M_n(ℂ) and D = diag(d₁,d₂,…,d_n). Then

(a): (DA)[i,:] = d_iA[i,:], for 1 ≤ i ≤ n, and
(b): (AD)[:,j] = d_jA[:,j], for 1 ≤ j ≤ n.

In particular, if D = αI is a scalar matrix, for some α ∈ ℂ, then DA = αA = AD.

3.

If x =

∈ M_n,1(ℂ) then x^*x = ∑ _i=1ⁿ|x_i|².

4.

Let A be an upper triangular matrix. If A^*A = AA^* then prove that A is a diagonal matrix. The same holds for lower triangular matrix.

5.

Let x =

,y =

∈ M_n,1(ℂ). Then, prove that xy^* = ⌊ ⌋
x1y1 x1y2 ⋅⋅⋅ x1yn-
| . . . |
|⌈ .. .. ⋅⋅⋅ .. |⌉
x y- x y- ⋅⋅⋅ x y--
n 1 n 2 n n

and y^*x = ∑ _i=1ⁿy_ix_i.

Definition 1.2.12. Two square matrices A and B are said to commute if AB = BA.

PICT PICT DRAFT Remark 1.2.13. Note that if A is a square matrix of order n and if B is a scalar matrix of order n then AB = BA. In general, the matrix product is not commutative. For example, consider A = [ ]
1 1
0 0 and B = [ ]
1 0
1 0 . Then, verify that AB = [ ]
2 0
0 0 ≠ [ ]
1 1
1 1 = BA.

Theorem 1.2.14. Suppose that the matrices A,B and C are so chosen that the matrix multiplications are defined.

1.: Then, (AB)C = A(BC). That is, the matrix multiplication is associative.
2.: For any k ∈ ℝ,(kA)B = k(AB) = A(kB).
3.: Then, A(B + C) = AB + AC. That is, multiplication distributes over addition.
4.: If A ∈ M_n(ℂ) then AI_n = I_nA = A.

Proof. Part 1. Let A = [a_ij] ∈ M_m,n(ℂ),B = [b_ij] ∈ M_n,p(ℂ) and C = [c_ij] ∈ M_p,q(ℂ). Then,

Using a similar argument, the next part follows. The other parts are left for the reader. _

Exercise 1.2.15.

1.

Let L₁,L₂ ∈ M_n(ℂ) be lower triangular matrices and U₁,U₂ ∈ M_n(ℂ) be upper triangular matrices. Further, let D ∈ M_n(ℂ) be a diagonal matrix. Then, prove that

(a): L₁L₂ is a lower triangular matrix.
(b): U₁U₂ is an upper triangular matrix.
(c): DL₁ and L₁D are lower triangular matrices.
(d): DU₁ and U₁D are upper triangular matrices.

2.

Let A ∈ M_m,n(ℂ). If Ax = 0 for all x ∈ M_n,1(ℂ) then prove that A = 0, the zero matrix.

3.

Let A,B ∈ M_m,n(ℂ). If Ax = Bx, for all x ∈ M_n,1(ℂ) then prove that A = B.

4.

Let A and B be two matrices such that the matrix product AB is defined.

(a): Prove that (AB)^* = B^*A^*.
(b): If A[1,:] = 0^* then (AB)[1,:] = 0^*. DRAFT
(c): If B[:,1] = 0 then (AB)[:,1] = 0.
(d): If A[i,:] = A[j,:] for some i and j then (AB)[i,:] = (AB)[j,:].
(e): If B[:,i] = B[:,j] for some i and j then (AB)[:,i] = (AB)[:,j].

5.

Construct matrices A and B, different from the one given earlier, that satisfy the following statements.

(a): The product AB is defined but BA is not defined.
(b): The products AB and BA are defined but they have different orders.
(c): The products AB and BA are defined, they have the same order but AB≠BA.
(d): Let A = and B = . Guess a formula for Aⁿ and Bⁿ and prove it?
(e): Let A = , B = and C = . Is it true that A²-2A+I = 0? What is B³ - 3B² + 3B - I? Is C³ = 3C²?

__________________________________

6.

Let A and B be two m × n matrices. Then, prove that (A + B)^* = A^* + B^*.

7.

Find a 2 × 2 nonzero matrix A satisfying A² = 0.

8.

Find a 2 × 2 nonzero matrix A satisfying A² = A and A≠I₂.

9.

Find 2 × 2 nonzero matrices A,B and C satisfying AB = AC but B≠C. That is, the cancelation law doesn’t hold.

10.

Let S =

and T =

. Then, determine the smallest positive integers m,n such that S^m = I and Tⁿ = I. PICT

DRAFT

11.

Let A =

. Compute A² and A³. Is A³ = I? Determine aA³ + bA + cA².

12.

Let A =

and B =

. Compute

(a): A - A^*,A + A^*,(3AB)^*- 4B^*A and 3A - 2A^*.
(b): (AB)[1,:],(AB)[3,:],(AB)[:,1] and (AB)[:,2].
(c): (B^*A^*)[:,1],(B^*A^*)[:,3],(B^*A^*)[1,:] and (B^*A^*)[2,:].

13.

Let a,b and c be indeterminate. Then, can we find A with complex entries satisfying A ⌊ ⌋
a
|⌈ b|⌉
c

? What if A [a ]

b

? Give reasons for your answer.

1.2.2 Inverse of a Matrix

Definition 1.2.16. Let A ∈ M_n(ℂ).

1.: Then, a square matrix B is said to be a left inverse of A, if BA = I_n.
2.: Then, a square matrix C is called a right inverse of A, if AC = I_n.
3.: Then, A is said to be invertible (or is said to have an inverse) if there exists a matrix B such that AB = BA = I_n.

PICT PICT DRAFT Lemma 1.2.17. Let A ∈ M_n(ℂ). If that there exist B,C ∈ M_n(ℂ) such that AB = I_n and CA = I_n then B = C.

Remark 1.2.18. Lemma 1.2.17 implies that whenever A is invertible, the inverse is unique. Thus, we denote the inverse of A by A^-1. That is, AA^-1 = A^-1A = I.

Example 1.2.19.

1.

Let A =

(a): If ad - bc≠0. Then, verify that A^-1 = .
(b): In particular, the inverse of equals .
(c): If ad - bc = 0 then prove that either A[1,:] = 0^* or A[:,1] = 0 or A[2,:] = αA[1,:] or A[:,2] = αA[:,1] for some α ∈ ℂ. Hence, prove that A is not invertible.
(d): Matrices , and do not have inverses. Justify your answer.

2.

Let A =

. Then, A^-1 = ⌊ ⌋
- 2 0 1
|⌈ 0 3 - 2|⌉

1 - 2 1

3.

Prove that the matrices A = ⌊ ⌋
1 1 1
|⌈1 1 1|⌉

1 1 1

and B =

are not invertible.

Solution: Suppose there exists C such that CA = AC = I. Then, using matrix product PICT PICT DRAFT

A[1,:]C = (AC )[1,:] = I[1,:] = [1,0,0] and A [2,:]C = (AC )[2,:] = I[2,:] = [0,1,0].

But A[1,:] = A[2,:] and thus [1,0,0] = [0,1,0], a contradiction.

Similarly, if there exists D such that BD = DB = I then

DB [:,1] = (DB )[:,1 ] = I[:,1],DB [:,2] = (DB )[:,2] = I[:,2] and DB [:,3] = I[:,3].

But B[:,3] = B[:,1] + B[:,2] and hence I[:,3] = I[:,1] + I[:,2], a contradiction.

Theorem 1.2.20. Let A and B be two invertible matrices. Then,

1.: (A^-1)^-1 = A.
2.: (AB)^-1 = B^-1A^-1.
3.: (A^*)^-1 = (A^-1)^*.

Proof. Part 1. Let B = A^-1 be the inverse of A. Then, AB = BA = I. Thus, by definition, B is invertible and B^-1 = A. Or equivalently, (A^-1)^-1 = A.

Part 3. As AA^-1 = A^-1A = I, we get (AA^-1)^* = (A^-1A)^* = I^*. Or equivalently, (A^-1)^*A^* = A^*(A^-1)^* = I. Thus, by definition (A^*)^-1 = (A^-1)^*. _ PICT

DRAFT

We will again come back to the study of invertible matrices in Sections 2.2 and 2.3.1.

Exercise 1.2.21.

1.

Let A be an invertible matrix. Then, prove that (A^-1)^r = A^-r, for all integers r.

2.

Find the inverse of [ ]
cos(θ) sin(θ)
sin (θ) - cos(θ)

and

3.

Let A₁,…,A_r be invertible matrices. Then, prove that the matrix B = A₁A₂ ⋅⋅⋅

A_r is also invertible.

4.

Let A ∈ M_n(ℂ) be an invertible matrix. Then, prove that

(a): A[i,:]≠0^T, for any i.
(b): A[:,j]≠0, for any j.
(c): A[i,:]≠A[j,:], for any i and j.
(d): A[:,i]≠A[:,j], for any i and j.
(e): A[3,:]≠αA[1,:] + βA[2,:], for any α,β ∈ ℂ, whenever n ≥ 3.
(f): A[:,3]≠αA[:,1] + βA[:,2], for any α,β ∈ ℂ, whenever n ≥ 3.

____________________

5.

Let x^* = [1 + i,2,3] and y^* = [2,-1 + i,4]. Prove that y^*x is invertible but yx^* is not invertible.

6.

Determine A that satisfies (I + 3A)^-1 = [ ]
1 2

2 1

7.

Determine A that satisfies (I - A)^-1 = ⌊ ⌋
- 2 0 1
| 0 3 - 2|
⌈ ⌉
1 - 2 1

. [See Example 1.2.19.2].

8.

Let A be a square matrix satisfying A³ + A - 2I = 0. Prove that A^-1 = 1
2

DRAFT

9.

Let A = [a_ij] be an invertible matrix. If B = [p^i-ja_ij], for some p ∈ ℂ, p≠0 then relate A^-1 and B^-1.

1.3 Some More Special Matrices

Definition 1.3.1.

1.

For 1 ≤ k ≤ m and 1 ≤ ℓ ≤ n, define a matrix

_kℓ ∈ M_m,n(ℂ) by (

_kℓ)_ij = {
1, if (k,ℓ) = (i,j)

0, otherwise.

Then, the matrices

_kℓ, for 1 ≤ k ≤ m and 1 ≤ ℓ ≤ n are called the standard basis elements for M_m,n(ℂ).

So, if _kℓ ∈ M_2,3(ℂ) then ₁₁ = [1 0 0 ]

0 0 0 = [1]

0 [ ]
1 0 0 ,₁₂ = [0 1 0]

0 0 0 = [ ]
1

0 [ ]
0 1 0 and ₂₂ = [ ]
0 0 0

0 1 0 = [ ]
0

1 .

2.

Let A ∈ M_m,n(ℝ).

(a): Then, A is called symmetric if A^T = A. For example, A = .
(b): Then, A is called skew-symmetric if A^T = -A. For example, A = .
(c): Then, A is called orthogonal if AA^T = A^TA = I. For example, A = .
(d): Let A ∈ M_n(ℂ). Then, A is said to be a permutation matrix if A has exactly one non-zero entry, namely 1, in each row and column. For example, I_n, for each positive integer n, , , and are permutation matrices. Verify that permutation matrices are Orthogonal matrices.

3.

Let A ∈ M_m,n(ℂ). PICT

DRAFT

(a): Then, A is called normal if A^*A = AA^*. For example, is a normal matrix.
(b): Then, A is called Hermitian if A^* = A. For example, A = .
(c): Then, A is called skew-Hermitian if A^* = -A. For example, A = .
(d): Then, A is called unitary if AA^* = A^*A = I. For example, A = .
Verify that Hermitian, skew-Hermitian and Unitary matrices are normal matrices.

4.

Then, A is called idempotent if A² = A. For example, A = [ ]
1 0
1 0

is idempotent.

5.

A vector u ∈ M_n,1(ℂ) such that u^*u = 1 is called a unit vector.

6.

A matrix that is symmetric and idempotent is called a projection matrix. For example, let u ∈ M_n,1(ℝ) be a unit vector. Then, A = uu^T is a symmetric and an idempotent matrix. Hence, A is a projection matrix. In particular, let u = √1-
5

[1,2]^T and A = uu^T. Then, u^Tu = 1 and for any vector x = [x₁,x₂]^T ∈ M_2,1(ℝ) note that

[ ]T Ax = (uuT )x = u (uTx) = x1-+√-2x2u = x1 +-2x2-, 2x1 +-4x2 . 5 5 5

Thus, Ax is the foot of the perpendicular from the point x on the vector [12]^T.

7.

Fix a unit vector a ∈ M_n,1(ℝ) and let A = 2aa^T - I_n. Then, verify that A ∈ M_n(ℝ) and Ay = 2(a^Ty)a - y, for all y ∈ ℝⁿ. This matrix is called the reflection matrix about the line containing the points 0 and a.

8.

Let A ∈ M_n(ℂ). Then, A is said to be nilpotent if there exists a positive integer n such that Aⁿ = 0. The least positive integer k for which A^k = 0 is called the order of nilpotency. For example, if A = [a_ij] ∈ M_n(ℂ) with a_ij equal to 1 if i-j = 1 and 0, otherwise then Aⁿ = 0 and A^ℓ≠0 for 1 ≤ ℓ ≤ n - 1.

DRAFT

Exercise 1.3.2.

1.

Consider the matrices

_ij ∈ M_n(ℂ) for 1 ≤ i,j,≤ n. Is

₁₂

₁₁ =

₁₁

₁₂? What about

₁₂

₂₂ and

₂₂

₁₂?

2.

Let {u₁,u₂,u₃} be three vectors in ℝ³ such that u_i^*u_i = 1, for 1 ≤ i ≤ 3, and u_i^*u_j = 0 whenever i≠j. Then prove the following.

(a): Let U = [u₁u₂u₃]. Then U^*U = I.
(b): If A = u_iu_i^*, for 1 ≤ i ≤ 3 then A² = A. Is A symmetric? Is A a projection matrix?
(c): If A = u_iu_i^* + u_ju_j^*, for i≠j then A² = A. Is A a projection matrix?

3.

Let A,B ∈ M_n(ℂ) be two unitary matrices. Then, prove that AB is also a unitary matrix.

4.

Let A ∈ M_n(ℂ) be a Hermitian matrix (A^* = A).

(a): Then, the diagonal entries of A are necessarily real numbers.
(b): For each B ∈ M_n(ℂ) prove that B^*AB is a Hermitian matrix.
(c): Further assume that A² = 0. Is it necessarily true that A = 0?
(d): Then, for any x ∈ M_n,1(ℂ), prove that x^*Ax is a real number.

5.

Let A ∈ M_n(ℂ). If x^*Ax is a real number for every x ∈ M_n,1(ℂ) then prove that A is a Hermitian matrix. [Hint: Use e_j,e_j + e_k and e_j + ie_k of M_n,1(ℂ) for x.]

6.

Let A and B be Hermitian matrices. Then, prove that AB is Hermitian if and only if AB = BA. PICT

DRAFT

7.

Let A ∈ M_n(ℂ) be a skew-Hermitian matrix. Then prove that

(a): the diagonal entries of A are either zero or purely imaginary.
(b): for each B ∈ M_n(ℂ) prove that B^*AB is a skew-Hermitian matrix.

8.

Let A be a complex square matrix. Then S₁ = 1
2

(A + A^*) is Hermitian, S₂ = 1
2

(A - A^*) is skew-Hermitian, and A = S₁ + S₂.

9.

Let A,B be skew-Hermitian matrices with AB = BA. Is the matrix AB Hermitian or skew-Hermitian?

10.

Let A be a nilpotent matrix. Prove that there exists a matrix B such that B(I + A) = I = (I + A)B. [If A^k = 0 then look at I - A + A² - ⋅⋅⋅

+ (-1)^k-1A^k-1].

11.

Are the matrices ⌊ ⌋
1 0 0
|⌈ 0 cosθ - sin θ|⌉

0 sin θ cos θ

and

orthogonal, for θ ∈ [-π,π)?

1.3.1 Submatrix of a Matrix

Definition 1.3.3. For a positive integer k, let us denote [k] = {1,…,k}. Also, let A ∈ M_m×n(ℂ).

1.: Then, a matrix obtained by deleting some of the rows and/or columns of A is said to be a submatrix of A.
2.: If S ⊆ [m] and T ⊆ [n] then by A(S|T) , we denote the submatrix obtained from A by deleting the rows with indices in S and columns with indices in T. By A[S,T], we mean A(S^c|T^c), where S^c is the complement of S in [m] and T^c is the complement of T in [n]. Whenever, S or T consist of a single element, then we just write the element. If S = [m], then we write A[S,T] = A[:,T] and if T = [n] then A[S,T] = A[S,:] which matches with our notation in Definition 1.1.1. DRAFT
3.: If m = n, the submatrix A[S,S] is called a principal submatrix of A.

Example 1.3.4.

1.: Let A = . Then, A[{1,2},{1,3}] = A[:,{1,3}] = , A[1,1] = [1],A[2,3] = [2],A[{1,2},1] = A[:,1] = ,A[1,{1,3}] = [15] and A are a few submatrices of A. But the matrices and are not submatrices of A.
2.: Take A = , S = {1,3} and T = {2,3}. Then, A[S,S] = ,A[T,T] = ,A(S∣S) = and A(T∣T) = are principal submatrices of A.

Let A be an n × m matrix and B be an m × p matrix. Suppose r < m. Then, we can decompose the matrices A and B as A = [PQ] and B = [H ]

K

, where P has order n×r and H has order r ×p. That is, the matrices P and Q are submatrices of A and P consists of the first r columns of A and Q consists of the last m - r columns of A. Similarly, H and K are submatrices of B and H consists of the first r rows of B and K consists of the last m - r rows of B. We now prove the following important theorem.

Theorem 1.3.5. Let A = [a_ij] = [PQ] and B = [b_ij] = [ ]
H
K be defined as above. Then,

DRAFT

Proof. The matrix products PH and QK are valid as the order of the matrices P,H,Q and K are respectively, n×r,r ×p,n× (m-r) and (m-r) ×p. Also, the matrices PH and QK are of the same order and hence their sum is justified. Now, let P = [P_ij],Q = [Q_ij],H = [H_ij], and K = [K_ij]. Then, for 1 ≤ i ≤ n and 1 ≤ j ≤ p, we have

Remark 1.3.6. Theorem 1.3.5 is very useful due to the following reasons:

1.: The order of the matrices P,Q,H and K are smaller than that of A or B.
2.: The matrices P,Q,H and K can be further partitioned so as to form blocks that are either identity or zero or matrices that have nice forms. This partition may be quite useful during different matrix operations.
3.: If we want to prove results using induction then after proving the initial step, one assumes the result for all r×r submatrices and then try to prove it for (r+1)×(r+1) submatrices.

Exercise 1.3.7.

1.

Complete the proofs of Theorems 1.2.5 and 1.2.14.

2.

Let x =

, y =

, A =

and B =

(a)

Then, prove that y = Ax gives the counter-clockwise rotation through an angle α.

(b)

Then, prove that y = Bx gives the reflection about the line y = tan(θ)x.

(c)

Let α = θ and compute y = (AB)x and y = (BA)x. Do they correspond to reflection? If yes, then about which line(s)?

(d)

Further, if y = Cx gives the counter-clockwise rotation through β and y = Dx gives the reflections about the line y = tan(δ)x.

i.: Then, prove that AC = CA and y = (AC)x gives the counter-clockwise rotation through α + β.
ii.: Then, prove that y = (BD)x and y = (DB)x give rotations. Which angles do they represent?

DRAFT

3.

Let A be an n × n matrix such that AB = BA for all n × n matrices B. Then, prove that A is a scalar matrix. That is, A = αI for some α ∈ ℂ (use matrices in Definition 1.3.1.1).

4.

Consider the two coordinate transformations

x₁	= a₁₁y₁ + a₁₂y₂
x₂	= a₂₁y₁ + a₂₂y₂

and

y₁	= b₁₁z₁ + b₁₂z₂
y₂	= b₂₁z₁ + b₂₂z₂

(a): Compose the two transformations to express x₁,x₂ in terms of z₁,z₂.
(b): Does the composition of two transformations obtained in the previous part correspond to multiplying two matrices? Give reasons for your answer.

5.

For A_n×n = [a_ij], the trace of A, denoted tr(A), is defined by tr(A) = a₁₁ + a₂₂ + ⋅⋅⋅

+ a_nn.

(a)

Compute tr(A) for A = [ ]
3 2

2 2

and A =

(b)

Let A be a matrix with A [ ]
1
2

= 2

and A

= 3

. If B =

then compute tr(AB).

(c)

Let A and B be two square matrices of the same order. Then, prove that

i.: tr(A + B) = tr(A) + tr(B).
ii.: tr(AB) = tr(BA).

(d)

Prove that there doesn’t exist matrices A,B ∈ M_n(ℂ) such that AB - BA = cI, for any c≠0.

6.

Let J ∈ M_n(ℝ) be a matrix having each entry 1.

(a): Then, verify that J² = nJ.
(b): Also, for any α₁,α₂,β₁,β₂ ∈ ℝ, verify that there exist α₃,β₃ ∈ ℝ such that DRAFT $(α1In + β1J )⋅(α2In + β2J) = α3In + β3J.$
(c): Let α,β ∈ ℝ such that α≠0 and α + nβ≠0. Now, define A = αI_n + βJ. Then, use the above to prove that A is invertible.

_________________________________________

7.

Let A =

(a): Find a matrix B such that AB = I₂.
(b): What can you say about the number of such matrices? Give reasons for your answer.
(c): Does there exist a matrix C such that CA = I₃? Give reasons for your answer.

8.

Let A =

⌊ | ⌋
1 0 |0 1
| | |
|| -0--1-|1--1-||
|⌈ 0 1 |1 0 |⌉
|
0 1 |0 1

and B =

⌊ | ⌋
1 2 |2 1
| | |
||--1---1-|2---1-||
|⌈ 1 1 |1 1 |⌉
|
- 1 1 |- 1 1

. Compute the matrix product AB using the block matrix multiplication.

9.

Let A =

. If P,Q and R are Hermitian, is the matrix A Hermitian?

10.

Let A =

, where A₁₁ is an n × n invertible matrix and c ∈ ℂ.

(a): If p = c - y^*A₁₁^-1x is nonzero, then verify that $[ ] [ ] A -111 0 1- A -111x [ * - 1 ] B = 0 0 + p - 1 y A11 - 1$ is the inverse of A.
(b): Use the above to find the inverse of and .

11.

Let x ∈ M_n,1(ℝ) be a unit vector.

(a): Define A = I_n - 2xx^T. Prove that A is symmetric and A² = I. The matrix A is commonly known as the Householder matrix.
(b): Let α≠1 be a real number and define A = I_n - αxx^T. Prove that A is symmetric and invertible. [The inverse is also of the form I_n + βxx^T, for some β.]

12.

Let A ∈ M_n(ℝ) be an invertible matrix and let x,y ∈ M_n,1(ℝ). Also, let β ∈ ℝ such that α = 1 + βy^TA^-1x≠0. Then, verify the famous Shermon-Morrison formula

(A + βxyT )-1 = A -1 - βA -1xyT A- 1. α

This formula gives the information about the inverse when an invertible matrix is modified by a rank (see Definition 2.2.26) one matrix.

13.

Suppose the matrices B and C are invertible and the involved partitioned products are defined, then verify that that

[ ] -1 [ ] A B 0 C -1 C 0 = B- 1 - B -1AC -1 .

14.

Let A be an m × n matrix. Then, a matrix G of order n×m is called a generalized inverse (for short, g-inverse) of A if AGA = A. For example, a generalized inverse of the matrix A = [1,2] is a matrix G = [ ]
1- 2α
α

, for all α ∈ ℝ. A generalized inverse G is called a pseudo inverse or a Moore-Penrose inverse if GAG = G and the matrices AG and GA are symmetric. Check that for α = 2-
5

the matrix G is a pseudo inverse of A. Further, among all the g-inverses, the inverse with the least euclidean norm also has α = 2-
5

1.4 Summary

In this chapter, we started with the definition of a matrix and came across lots of examples. We recall these examples as they will be used in later chapters to relate different ideas:

We also learnt product of two matrices. Even though it seemed complicated, it basically tells that multiplying by a matrix on the