Matrices (2nd Sem)

Chapter

Matrices

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Matrices :

A set of mn numbers (real or complex) arranged in the form of a rectangular array having m rows and n columns is called an m x n matrix [to be read as ‘m by n’ matrix].

An m x n matrix is usually written as:

$$A = \begin{bmatrix} a_{11} & a_{12} & \dots & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & \dots & a_{2n} \\ a_{31} & a_{32} & \dots & \dots & a_{3n} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \dots & \dots & a_{mn} \end{bmatrix}$$

In a compact form, the above matrix is represented by:

$$A=[a_{ij}]i=1,2,3,\dots ,m,j=1,2,3,\dots ,n$$

or simply by

$$[a_{ij}{]}_{m\times \mathrm{n}}$$

We write the general element of the matrix and enclose it in brackets of type [ ] or of the type ( ).

(i) Square Matrices

Definition: An $m \times n$ matrix for which $m = n$ (i.e., the number of rows is equal to the number of columns) is called a square matrix of order $n$. It is also called an $n$-rowed square matrix. The elements $a_{ij}$ of a square matrix $A = [a_{ij}]_{n \times m}$ for which $i = j$.

$$A = \begin{bmatrix} 1 & 2 & 5 \\ 1 & 3 & 3 \\ 3 & 4 & 1 \end{bmatrix}_{3 \times 3}$$

(ii) Unit Matrix or Identity Matrix

Definition: A square matrix, each of whose diagonal elements is 1 and whose non-diagonal elements are 0, is called a Unit Matrix or an Identity Matrix and is denoted by I.
$I_n$ will denote a unit matrix of order $n$.

Thus, a square matrix $A = [a_{ij}]$ is a unit matrix if

$$a_{ij}=1\text{when }i=j,\text{and}{a}_{ij}=0\text{when }i\mathrm{\ne }j\mathrm{.}$$$$I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ $$I_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

(iii) Null Matrix:

Definition: The $m \times n$ matrix whose elements are all 0 is called the null matrix (or zero matrix) of the type $m \times n$. It is usually denoted by O or more clearly by $O_{m,n}$. Often, a null matrix is simply denoted by the symbol 0 read as ‘zero’.

$$O_{3 \times 3} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \quad O_{3 \times 2} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$$

(iv) Row Matrix and Column Matrix

Definition: Any $1 \times n$ matrix, which has only one row and $n$ columns, is called a row matrix or row vector.
Similarly, any $m \times 1$ matrix, which has $m$ rows and only one column, is a column matrix or a column vector.

$$Y = \begin{bmatrix} 2 \\ 5 \\ 4 \end{bmatrix}_{3 \times 1} \quad X = \begin{bmatrix} 4 & 7 & 1 \end{bmatrix}_{1 \times 3}$$

Equality of Two Matrices

Definition: Two matrices $A = [a_{ij}]$ and $B = [b_{ji}]$ are said to be equal if:

1. They are the same size, and
2. The elements in the corresponding places of the two matrices are the same, i.e., $$a_{ij}=b_{ij}\text{ for each pair of subscripts }i\text{ and }j\mathrm{.}$$

If two matrices A and B are equal, we write A = B.
If two matrices A and B are not equal, we write A ≠ B.

If two matrices are not of the same size, they cannot be equal.

Addition of Matrices

Definition: The sum of two matrices $A$ and $B$ of the same order is obtained by adding their corresponding elements.

If $A = [a_{ij}]$ and $B = [b_{ij}]$ are two matrices of the same order $m \times n$, then their sum is given by:

$$A+B=[a_{ij}+b_{ij}]$$

for all values of $i$ and $j$.

Conditions for Addition:

• The two matrices must have the same order (same number of rows and columns).
• The addition is performed element-wise.

Example:

Let

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$$

Then,

$$A+B=\left[\begin{array}{cc}1+5& 2+6\\ 3+7& 4+8\end{array}\right]=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]$$

Multiplication of Two Matrices

Definition: The product of two matrices $A$ and $B$ is defined if the number of columns of the first matrix $A$ is equal to the number of rows of the second matrix $B$.

If $A$ is of order $m \times n$ and $B$ is of order $n \times p$, then their product $AB$  is a matrix of order $m \times p$, where each element is calculated as:

$$(AB{)}_{ij}=\sum _{k=1}^n{a}_{ik}{b}_{k\mathrm{j}}$$

Conditions for Multiplication:

1. The number of columns of the first matrix must be equal to the number of rows of the second matrix.
2. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Example:

Let

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$$

Now, calculating $AB$:

$$AB = \begin{bmatrix} (1 \times 5 + 2 \times 7) & (1 \times 6 + 2 \times 8) \\ (3 \times 5 + 4 \times 7) & (3 \times 6 + 4 \times 8) \end{bmatrix}$$
$$= \begin{bmatrix} (5+14) & (6+16) \\ (15+28) & (18+32) \end{bmatrix}$$
$$=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$$

Key Points:

• Matrix multiplication is NOT commutative, i.e., $AB \neq BA$ in general.
• Associative Property: $(AB)C = A(BC)$.
• Distributive Property: $A(B+C) = AB + AC$.

Triangular, Diagonal and Scalar Matrices

(i) Upper Triangular Matrix

Definition: A square matrix $A = [a_{ij}]$ is called an upper triangular matrix if $a_{ij} = 0$ whenever $i > j$.

$$\begin{bmatrix} 1 & 3 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 7 \end{bmatrix}$$

(ii) Lower Triangular Matrix

Definition: A square matrix $A = [a_{ij}]$ is called a lower triangular matrix if $a_{ij} = 0$ whenever $i < j$.

$$\begin{bmatrix} 1 & 0 & 0 \\ 2 & 2 & 0 \\ 4 & 7 & 6 \end{bmatrix}$$

Lecture 1

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Transpose of a Matrix

Definition:

Let $A = [a_{ij}]_{m \times n}$, then the $n \times m$ matrix obtained from $A$ by changing its rows into columns and its columns into rows is called the transpose of $A$ and is denoted by the symbol $A'$ or $A^T$.

$$A = \begin{bmatrix} 1 & 2 & 6 \\ 3 & 4 & 7 \\ 5 & 2 & 8 \end{bmatrix}$$ $$A^T = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 2 \\ 6 & 7 & 8 \end{bmatrix}$$

Theorems:

If $A'$ and $B'$ be the transposes of matrices $A$ and $B$ respectively, then:

1. $(A')' = A$
   
2. $(A + B)' = A' + B'$
   
3. $(kA)' = kA'$, where $k$ is any complex number.
4. $(AB)' = B'A'$, where $A$ and $B$ are conformable for multiplication.

Orthogonal Matrix

Definition:

A square matrix $A$ is said to be Orthogonal if:

$$A^T A = I$$

where $A^T$ is the transpose of $A$ and $I$ is the identity matrix.

Theorem 1

If $A$ and $B$ are $n$-rowed orthogonal matrices, then the products $AB$ and $BA$ are also orthogonal matrices.

Theorem 2

If $A$ is an orthogonal matrix, then its transpose $A'$ and its inverse $A^{-1}$ are also orthogonal

Lecture 2

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Q1: Determine the values of $\alpha, \beta, \gamma$ when is orthogonal.

$$A=\left[\begin{array}{ccc}0& 2\beta & \gamma \\ \alpha & \beta & -\gamma \\ \alpha & -\beta & \gamma \end{array}\right]$$

$$<b><span\; style="font-size:\; medium;">Answer\; :\; </span>For\; the\; solution,\; see\; the\; video\; given\; below\; (हल\; के\; लिए,\; नीचे\; दी\; गई\; विडियो\; देखे)</b>$$

Q2: Show that the matrix is orthogonal.

$$A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$$

$$<br>$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Q3: Show that the matrix is orthogonal.

$$A = \frac{1}{3} \begin{bmatrix} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{bmatrix}$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)​

Q4: Verify that the matrix is orthogonal.

$$A = \frac{1}{3} \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ -2 & 2 & -1 \end{bmatrix}$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)​

Lecture 3

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Conjugate of a Matrix

If I = √−1 , then z = x + iy is called a complex number where x and y are any real numbers.
If z = x + iy, then z̅ = x − iy is called the Conjugate of the complex number z.

We have z z̅ = (x + iy)(x - iy) = x² + y² i.e., is real

Also if z = z̅, then x + iy = x - iy i.e., 2iy = 0 i.e., y = 0 i.e., z is real.

Conversely, if z is real then z̅ = z

z = x + iy, then z̅ = x - iy (z̅̅) = x + iy = z

If z₁ and z₂ are two complex numbers, then it can be easily seen that

(i) z̅₁ + z₂ = z̅₁ + z̅₂
(ii) z̅₁ z₂ = (z̅₁) + (z̅₂)

Conjugate of a Matrix

Definition: The matrix obtained from any given matrix A on replacing its elements by the corresponding conjugate complex numbers is called the conjugate of A and is denoted by Ā.

Thus if A = [aᵢⱼ]ₘₓₙ, then Ā = [āᵢⱼ]ₘₓₙ, where āᵢⱼ denotes the conjugate complex of aᵢⱼ.

If A be a matrix over the field of real numbers, then obviously Ā coincides with A.

$$A = \begin{bmatrix} 2 + 3i & 4 - 7i & 8 \\ - i & 6 & 9 + i \end{bmatrix}$$
$$Ā = \begin{bmatrix} 2 - 3i & 4 + 7i & 8 \\ i & 6 & 9 - i \end{bmatrix}$$

Theorems:

If Ā and B̅ be the conjugates of A and B respectively, then

(i) (Ā̅) = A

(ii) (A + B)̅ = Ā + B̅, A and B being of the same size

(iii) (kA)̅ = k̅Ā, k is being any complex number

(iv) (AB)̅ = ĀB̅, A and B being conformable to multiplication

Transposed of Conjugate of a Matrix

Definition: The transpose of the conjugate of a matrix A is called the transposed conjugate of A and is denoted by Aᶿ or by A*.
Obviously, the conjugate of the transpose of A is the same as the transpose of the conjugate of A, i.e.,

(A̅′) = (A′)̅ = Aᶿ

If A = $[a_{ij}]_{m \times n}$, then Aᶿ = $[b_{ij}]_{n \times m}$

Where b₍ᵢⱼ₎ = ā₍ⱼᵢ₎, i.e., the $(j, i)^{th}$ element of Aᶿ = the conjugate complex of the $(i, j)^{th}$ element of A

$$A = \begin{bmatrix} 1 + 2i & 2 - 3i & 3 + 4i \\ 4 - 5i & 5 + 6i & 6 - 7i \\ 8 & 7 + 8i & 7 \end{bmatrix}$$ $$A′ = \begin{bmatrix} 1 + 2i & 4 - 5i & 8 \\ 2 - 3i & 5 + 6i & 7 + 8i \\ 3 + 4i & 6 - 7i & 7 \end{bmatrix}$$

Theorems:

If Aᶿ and Bᶿ be the transposed conjugates of A and B respectively, then

1. (Aᶿ)ᶿ = A
2. (A + B)ᶿ = Aᶿ + Bᶿ, where A and B are of the same size
3. (kA)ᶿ = kAᶿ, where k is any complex number
4. (AB)ᶿ = Bᶿ Aᶿ, where A and B are conformable to multiplication

Symmetric and Skew-symmetric Matrices

Symmetric Matrix (Definition):

A square matrix A = [aᵢⱼ] is said to be symmetric if its (i, j)ᵗʰ element is the same as its (j, i)ᵗʰ element, i.e., if aᵢⱼ = aⱼᵢ for all i, j.

Example Matrices:

$$\begin{bmatrix} 2 & 4 \\ 4 & 3 \end{bmatrix}$$
$$\begin{bmatrix} 1 & i & -2i \\ i & -2 & 4 \\ -2i & 4 & 3 \end{bmatrix}$$

Hermitian and Skew-Hermitian Matrices

Hermitian Matrix (Definition):

A square matrix A = [aᵢⱼ] is said to be Hermitian if its (i, j)ᵗʰ element of A is equal to the conjugate complex of the (j, i)ᵗʰ element of A, i.e., if

$$a_{ij}=\overline{a_{ji}}\text{ for all }i,j\mathrm{.}$$

Example Matrices:

$$\begin{bmatrix} a & b + ic \\ b - ic & d \end{bmatrix}$$
$$\begin{bmatrix} 1 & 2 - 3i & 3 + 4i \\ 2 + 3i & 0 & 4 - 5i \\ 3 - 4i & 4 + 5i & 2 \end{bmatrix}$$

Note:

If A is a Hermitian matrix, then

$$aᵢⱼ = \overline{aⱼᵢ}$$

By definition, aᵢⱼ is real for all i, thus every diagonal element of a Hermitian matrix must be real.

Skew-Hermitian Matrix (Definition):

A square matrix A = [aᵢⱼ] is said to be Skew-Hermitian if its (i, j)ᵗʰ element of A is equal to the negative of the conjugate complex of the (j, i)ᵗʰ element of A, i.e., if

$$a_{ij}=-\overline{a_{ji}}\text{ for all }i,j\mathrm{.}$$

Example Matrices:

$$\begin{bmatrix} 0 & -2 - i \\ 2 - i & 0 \end{bmatrix}$$
$$\begin{bmatrix} 0 & -i & 3 + 4i \\ i & 0 & -3 + 4i \\ -3 - 4i & 3 - 4i & 0 \end{bmatrix}$$

Properties of a Skew-Hermitian Matrix:

1. Symmetry Condition: $$aᵢⱼ = -\overline{aⱼᵢ} \text{ for all } i, j$$
   
2. Diagonal Elements:
   If A is a Skew-Hermitian matrix, then $$aᵢᵢ = -\overline{aᵢᵢ} \text{ (by definition)}$$
   $$aᵢᵢ + \overline{aᵢᵢ} = 0$$This implies that the diagonal elements of a Skew-Hermitian matrix must be purely imaginary or zero.

Note:

If A is a Skew-Hermitian matrix, then

$$aᵢⱼ = -\overline{aⱼᵢ}$$

By definition, aᵢⱼ must be either a pure imaginary number or zero.

Lecture 4

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Q1: Show that the matrix is Skew-Hermitian.

$$\begin{bmatrix} i & 3 + 2i & -2 - i \\ -3 + 2i & 0 & 3 - 4i \\ 2 - i & -3 - 4i & -2i \end{bmatrix}$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)​

Q2: Express as the sum of a Hermitian and a Skew-Hermitian matrix

$$\begin{bmatrix} -2 + 3i & 1 - i & 2 + i \\ 3 & 4 - 5i & 5 \\ 1 & 1 + i & -2 + 2i \end{bmatrix}$$

$$$$Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Q3: Show that the matrix is skew-symmetric

Q4: Show that the matrix A, where

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)​​​

Q5: If prove that A is a Hermitian matrix.

$$A = \begin{bmatrix} 3 & 2 - 3i & 3 + 5i \\ 2 + 3i & 5 & -i \\ 3 - 5i & -i & 7 \end{bmatrix}$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)​

Lecture 5

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Unitary Matrix

Definition: A square matrix $A$ is said to be unitary if $A^\theta A = I$.

Since $|A^\theta| = |A|$ and $|A^\theta A| = |A^\theta||A|$, therefore if $A^\theta A = I$, we have $|A| |A^\theta| = 1$.

Thus, the determinant of a unitary matrix is of unit modulus.

If $A$ is a unitary matrix, then $|A| |A^\theta| = 1$ and so $|A| \neq 0$, i.e., $$AA is non-singular and invertible.

Hence, $A^\theta A = I$ implies $A A^\theta = I$.

Thus, $A$ is a unitary matrix if and only if $A^\theta A = I = A A^\theta$.

Q1: Prove $B = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & 1 + i \\ 1 - i & -1 \end{bmatrix}$ is Unitary.

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Q2: Show that the matrix

$A = \begin{bmatrix} \alpha + i\gamma & -\beta + i\delta \\ \beta + i\delta & \alpha - i\gamma \end{bmatrix}$

is unitary if $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = 1$.

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Q3: Prove that the matrix $\begin{bmatrix} \frac{1 + i}{2} & \frac{-1 + i}{2} \\ \frac{1 + i}{2} & \frac{1 - i}{2} \end{bmatrix}$ is unitary.

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Lecture 6

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