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Unit 1 - Notes

MTH165 7 min read

Unit 1: Linear Algebra

1. Review of Matrices

Definition

A matrix is a rectangular array of numbers (real or complex) arranged in rows and columns, enclosed by brackets.

A = [a_{ij}]_{m \times n}

where

m

is the number of rows and

n

is the number of columns.

Important Types of Matrices

Square Matrix: Number of rows equals number of columns ( $m=n$ ).
Diagonal Matrix: A square matrix where all non-diagonal elements are zero ( $a_{ij} = 0$ for $i \neq j$ ).
Identity (Unit) Matrix ( $I_n$ ): A diagonal matrix where all diagonal elements are 1.
Null (Zero) Matrix ( $O$ ): All elements are zero.
Upper Triangular: All elements below the main diagonal are zero.
Lower Triangular: All elements above the main diagonal are zero.
Symmetric Matrix: $A^T = A$ (Elements are symmetric about the main diagonal).
Skew-Symmetric Matrix: $A^T = -A$ (Diagonal elements must be zero).
Orthogonal Matrix: $A A^T = A^T A = I$ . (Implying $A^T = A^{-1}$ ).

2. Elementary Operations of Matrices

Elementary operations are fundamental manipulations used to find the rank, inverse, or solve linear systems. They do not alter the order of the matrix.

Elementary Row Operations (ERO)

There are three valid operations:

Interchange: Interchanging any two rows ( $R_i \leftrightarrow R_j$ ).
Scaling: Multiplying a row by a non-zero constant ( $R_i \rightarrow kR_i$ ).
Addition: Adding a multiple of one row to another ( $R_i \rightarrow R_i + kR_j$ ).

Note: Similar operations apply to columns (Column Operations), but row operations are standard for Gaussian elimination.

3. Rank of a Matrix

Definition

The rank of a matrix $A$ , denoted as $\rho(A)$ or $r(A)$ , is the order of the largest non-vanishing (non-zero) minor of the matrix.

Minor: The determinant of a square sub-matrix.
If $\rho(A) = r$ , then at least one minor of order $r$ is non-zero, and every minor of order $r+1$ is zero.

Methods to Find Rank

1. Echelon Form Method (Recommended)

Transform the matrix using EROs to an Upper Triangular form.

All zero rows must be at the bottom.
The number of leading zeros in a lower row is greater than the row above it.
Result: $\rho(A) =$ Number of non-zero rows in Echelon form.

2. Normal Form (Canonical Form)

Reduce the matrix using both row and column operations to the form:

\begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix}

Here,

I_r

is the identity matrix of order

r

Result: $\rho(A) = r$ .

4. Linear Dependence and Independence of Vectors

Consider a set of vectors $X_1, X_2, \dots, X_n$ .
Consider the linear combination:

c_1X_1 + c_2X_2 + \dots + c_nX_n = 0

where

c_i

are scalars.

Linear Independence

The vectors are Linearly Independent if the equation above holds only when all scalars are zero:

c_1 = c_2 = \dots = c_n = 0

Linear Dependence

The vectors are Linearly Dependent if there exists a set of scalars, not all zero, such that the relation holds. One vector can be expressed as a linear combination of the others.

Testing Dependence via Matrices

Form a matrix where the vectors are rows (or columns).

Calculate the determinant (if square). If $|A| = 0$ , vectors are dependent. If $|A| \neq 0$ , vectors are independent.
Find the Rank.
- If $\rho(A) =$ number of vectors, they are Independent.
- If $\rho(A) <$ number of vectors, they are Dependent.

5. Solution of Linear System of Equations

A system of linear equations can be written in matrix form as:

AX = B

$A$ : Coefficient matrix
$X$ : Column matrix of variables (unknowns)
$B$ : Column matrix of constants

Case 1: Non-Homogeneous System ( $B \neq 0$ )

Form the Augmented Matrix $C = [A : B]$ .
Let $n$ be the number of unknowns.

Rouche-Capelli Theorem:

Inconsistent (No Solution):
$\rho(A) \neq \rho(A : B)$
Consistent (Solution exists):
- Unique Solution: If $r = n$ (Rank equals number of variables).
- Infinite Solutions: If $r < n$ . (The system has $n-r$ free variables/parameters).

Case 2: Homogeneous System ( $B = 0$ )

System is $AX = 0$ . This system is always consistent (has at least the trivial solution $X=0$ ).

Trivial Solution (Zero solution only):
If $\rho(A) = n$ (or $|A| \neq 0$ ).
Non-Trivial Solution (Infinite non-zero solutions):
If $\rho(A) < n$ (or $|A| = 0$ ).

6. Inverse of Matrices

The inverse of a square matrix $A$ exists only if $A$ is non-singular ( $|A| \neq 0$ ). It is denoted as $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$ .

Gauss-Jordan Method (Elementary Operation Method)

This is an algorithmic approach suitable for engineering computations.

Write the augmented matrix $[A : I]$ .
Apply Row Operations to reduce $A$ to the Identity matrix $I$ . The same operations apply to $I$ on the right side.
$[A : I] \xrightarrow{Row Operations} [I : A^{-1}]$
The matrix obtained on the right side is the inverse.

7. Eigenvalues and Eigenvectors

Definitions

Let $A$ be a square matrix of order $n$ .
If there exists a scalar $\lambda$ and a non-zero column vector $X$ such that:

AX = \lambda X

(A - \lambda I)X = 0

Then:

$\lambda$ is called the Eigenvalue (Characteristic root / Latent root).
$X$ is called the Eigenvector corresponding to $\lambda$ .

Characteristic Equation

To find eigenvalues, the system $(A - \lambda I)X = 0$ must have a non-trivial solution (since $X \neq 0$ ). This implies the determinant of the coefficient matrix must be zero:

|A - \lambda I| = 0

This polynomial equation in

\lambda

is called the Characteristic Equation.

Procedure

Form the Characteristic Equation $|A - \lambda I| = 0$ .
Solve the equation to find the roots $\lambda_1, \lambda_2, \dots$ . These are the eigenvalues.
For each eigenvalue $\lambda_i$ , substitute it back into $(A - \lambda_i I)X = 0$ and solve for $X$ using matrix row reduction.

8. Properties of Eigenvalues

These properties are essential for simplifying calculations. Let $A$ be a square matrix with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ .

Sum Property: Sum of eigenvalues equals the Trace of $A$ (Sum of main diagonal elements).
$\sum \lambda_i = \text{Trace}(A)$
Product Property: Product of eigenvalues equals the Determinant of $A$ .
$\prod \lambda_i = |A|$
Triangular/Diagonal Matrices: The eigenvalues of a triangular or diagonal matrix are exactly the diagonal elements themselves.
Inverse: If $\lambda$ is an eigenvalue of $A$ , then $1/\lambda$ is an eigenvalue of $A^{-1}$ (provided $|A| \neq 0$ ).
Power: If $\lambda$ is an eigenvalue of $A$ , then $\lambda^k$ is an eigenvalue of $A^k$ (where $k$ is a positive integer).
Scalar Multiplication: If $\lambda$ is an eigenvalue of $A$ , then $k\lambda$ is an eigenvalue of $kA$ .
Symmetric Matrices: The eigenvalues of a real symmetric matrix are always Real.
Similar Matrices: If $A$ and $P^{-1}AP$ are similar matrices, they have the same eigenvalues.

9. Cayley-Hamilton Theorem

Statement

"Every square matrix satisfies its own characteristic equation."

If the characteristic equation of a square matrix $A$ is:

a_0\lambda^n + a_1\lambda^{n-1} + \dots + a_n = 0

Then, replacing

\lambda

with the matrix

A

and the constant term with

a_nI

a_0A^n + a_1A^{n-1} + \dots + a_nI = 0

(Where $0$ represents the Null Matrix).

Applications of Cayley-Hamilton Theorem

To Find the Inverse ( $A^{-1}$ ):
Multiply the matrix equation by $A^{-1}$ :

$a_0A^{n-1} + a_1A^{n-2} + \dots + a_{n-1}I + a_nA^{-1} = 0$
$A^{-1} = -\frac{1}{a_n} [a_0A^{n-1} + \dots + a_{n-1}I]$
To Find Higher Powers ( $A^k$ ):
If you need to calculate a high power (e.g., $A^8$ ) without multiplying $A$ eight times, you can use the characteristic equation to express $A^n$ in terms of lower powers of $A$ .

Verification Steps

Find Characteristic Equation in terms of $\lambda$ .
Substitute $A$ for $\lambda$ in the equation.
Calculate powers of $A$ ( $A^2, A^3$ , etc.).
Perform matrix addition/subtraction according to the equation.
Verify the result is the Null Matrix.

Unit 2

Unit 1 - Notes

Table of Contents

Unit 1: Linear Algebra

1. Review of Matrices

Definition

Important Types of Matrices

2. Elementary Operations of Matrices

Elementary Row Operations (ERO)

3. Rank of a Matrix

Definition

Methods to Find Rank

1. Echelon Form Method (Recommended)

2. Normal Form (Canonical Form)

4. Linear Dependence and Independence of Vectors

Linear Independence

Linear Dependence

Testing Dependence via Matrices

5. Solution of Linear System of Equations

Case 1: Non-Homogeneous System ()

Case 2: Homogeneous System ()

6. Inverse of Matrices

Gauss-Jordan Method (Elementary Operation Method)

7. Eigenvalues and Eigenvectors

Definitions

Characteristic Equation

Procedure

8. Properties of Eigenvalues

9. Cayley-Hamilton Theorem

Statement

Applications of Cayley-Hamilton Theorem

Verification Steps

Case 1: Non-Homogeneous System ( $B \neq 0$ )

Case 2: Homogeneous System ( $B = 0$ )