Module 1: Systems of Linear Equations, Matrix Operations, and Quadratic Forms

Introduction to Matrices and Matrix Types

• Overview & Applications:

  • Information in science, business, and mathematics is organized into rows and columns forming rectangular arrays called matrices.

  • Matrix representations of systems of equations are central to computational algorithms because computers are designed to manipulate structured arrays of numbers.

  • Matrices are formal mathematical objects governed by linear algebra, possessing structural properties and application domains across engineering, data science, and physical modeling.

• Matrix Definition:

  • A matrix is a rectangular array of numbers enclosed in brackets.

  • A matrix containing $m$ rows and $n$ columns has order $m \times n$ (read "$m$ by $n$").

  • General matrix structure:

$A = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{pmatrix}_{m \times n}$

• Examples of Matrices:

  • Example 1:

$A = \begin{pmatrix} 0.3 & 1 & -5 \\ 0 & -0.2 & 16 \end{pmatrix}_{2 \times 3}$

• Example 2:

$B = \begin{pmatrix} 4 \\ 0.5 \end{pmatrix}_{2 \times 1}$

• Classification of Matrix Types:

  • Row Matrix: A matrix possessing only one row ($1 \times n$).

  • Example:

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

• Column Matrix: A matrix possessing only one column ($m \times 1$).

  • Example:

$A = \begin{pmatrix} -5 \\ 6 \\ 2 \end{pmatrix}$

• Null Matrix (Zero Matrix): A matrix in which every entry is equal to zero.

  • Example:

$A = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

• Square Matrix: A matrix with an equal number of rows and columns ($m = n$).

  • Example:

$A = \begin{pmatrix} 2 & -3 \\ 4 & 1 \end{pmatrix}$

• Diagonal Matrix: A square matrix in which all non-leading diagonal entries are zero ($a_{ij} = 0$ for all $i \neq j$).

  • Example:

$A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & -5 \end{pmatrix}$

• Scalar Matrix: A diagonal matrix whose leading diagonal entries are all identical.

  • Example:

$A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}$

• Identity Matrix (Unit Matrix): A scalar matrix whose leading diagonal entries are all equal to $1$.

  • Example:

$A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Matrix Operations

• Equality of Matrices:

  • Two matrices $A$ and $B$ are equal ($A = B$) if and only if both matrices have the same order and their corresponding elements are identical ($a_{ij} = b_{ij}$ for all $i, j$).

• Addition and Subtraction:

  • For matrices $A$ and $B$ of identical order $m \times n$, the sum/difference $A \pm B$ is an $m \times n$ matrix obtained by adding or subtracting corresponding entries.

• Scalar Multiplication:

  • For a scalar $k$ and matrix $A$, the product $kA$ is an $m \times n$ matrix produced by multiplying every individual entry of $A$ by $k$.

• Matrix Multiplication:

  • The product matrix $AB$ of matrix $A$ (order $m \times n$) and matrix $B$ (order $p \times q$) is defined if and only if the number of columns in $A$ equals the number of rows in $B$ ($n = p$).

  • The resulting product matrix $AB$ has dimensions $m \times q$.

Row Echelon Form and Reduced Row Echelon Form

• Row Echelon Form:

  • A matrix $A$ is in echelon form if it meets two fundamental conditions:

  1. All zero rows (rows consisting entirely of zeros), if any, are situated at the bottom of the matrix.

  2. Each leading nonzero entry (the first nonzero element in a row) is located strictly to the right of the leading nonzero entry of the preceding row.

  • Example:

$A = \begin{pmatrix} 2 & 3 & 2 & 0 & 4 & 5 & -6 \\ 0 & 0 & 0 & 1 & -3 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}$

• Reduced Row Echelon Form (Row Canonical Form):

  • A matrix $A$ is in row canonical form if it satisfies the echelon conditions (1) and (2) along with two additional requirements:

  1. Each pivot entry (leading nonzero entry) is equal to $1$.

  2. Each pivot entry is the unique nonzero entry in its containing column.

  • Example 1:

$A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$

• Example 2:

$B = \begin{pmatrix} 0 & 1 & 3 & 0 & 0 & 4 \\ 0 & 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 0 & 1 & 2 \end{pmatrix}$

Elementary Row Operations and Equivalent Matrices

• Elementary Row Operations:

  • Given a matrix $A$ with rows $R_1, R_2, \dots, R_m$, the permissible elementary row operations are:

  1. Row Interchange: Exchanging row $R_i$ with row $R_j$, written as R_i \longleftrightarrow R_j$.\n 2. **Row Scaling:** Replacing row R_i$with a non-zero scalar multiple$k R_i$, written as$R_i \longrightarrow k R_i$.

  2. Row Addition: Replacing row $R_j$ with the linear sum of $R_j$ and a scalar multiple $k R_i$ of row $R_i$, written as $R_j \longrightarrow R_j + k R_i$.

  • Note: The symbol "$\longrightarrow$" denotes "changes to".

• Equivalent Matrices:

  • Two matrices $A$ and $B$ are row equivalent if one can be transformed into the other through a finite sequence of elementary row operations.

• Row Equivalence Examples:

  • Example 1: Show that the following matrices are row equivalent:

$\begin{pmatrix} 1 & 2 & -3 & 1 & 2 \\ 2 & 4 & -4 & 6 & 10 \\ 3 & 6 & -6 & 9 & 13 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 1 & 2 & -3 & 1 & 2 \\ 0 & 0 & 2 & 4 & 6 \\ 0 & 0 & 0 & 0 & -2 \end{pmatrix}$

• Example 2: Show that the following matrices are row equivalent:

$\begin{pmatrix} 1 & 1 & -2 & 4 & 5 \\ 2 & 2 & -3 & 1 & 3 \\ 3 & 3 & -4 & -2 & 1 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 1 & 1 & 0 & -10 & -9 \\ 0 & 0 & 1 & -7 & -7 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$

Rank of a Matrix

• Formal Definitions:

  • Sub-matrix Definition: A matrix $A$ has rank $r$ if:

  1. It contains at least one square sub-matrix of order $r$ that is non-singular (determinant is non-zero).

  2. Every square sub-matrix of order greater than $r$ is singular (determinant equals zero).

  • Echelon Definition: The rank $r$ of matrix $A$ equals the exact number of non-zero rows in its canonical form or row echelon form.

  • Notation: $\text{rank}(A) = r$ or $\rho(A) = r$.

• Worked Problems and Applications:

  • Problem 1: Find the rank of the matrix:

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

• Problem 2 (Social Media Influence Analysis): A digital marketing agency analyzes interaction among four social media influencers represented by the matrix:

$\begin{pmatrix} 0 & 1 & -3 & -1 \\ 1 & 0 & 1 & 1 \\ 3 & 1 & 0 & 2 \\ 1 & 1 & -2 & 0 \end{pmatrix}$

* Positive entries indicate positive influence, negative entries indicate opposing influence, and zero indicates no direct influence.
* Task: Determine the matrix rank and state whether the interaction data provides four independent influence patterns or if some information is redundant.

• Problem 3: Find the rank of the matrix:

$\begin{pmatrix} 1 & 2 & 3 & 0 \\ 2 & 4 & 3 & 2 \\ 3 & 2 & 1 & 3 \\ 6 & 8 & 7 & 5 \end{pmatrix}$

• Problem 4 (Machine Learning Feature Selection): A machine learning engineer analyzes relationship metrics among four input features for customer behavior prediction using the matrix:

$\begin{pmatrix} 1 & 2 & -1 & 4 \\ 2 & 4 & 3 & 4 \\ 1 & 2 & 3 & 4 \\ -1 & -2 & 6 & -7 \end{pmatrix}$

* Task: Determine the matrix rank and conclude whether all four features contribute independent information to the predictive model or whether redundant features exist.

• Problem 5: Find the rank of the matrix:

$\begin{pmatrix} 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \\ 11 & 12 & 13 & 14 \\ 16 & 17 & 18 & 19 \end{pmatrix}$

Systems of Linear Equations

• General Form:

  • A linear system of $m$ equations in $n$ unknowns $x_1, x_2, \dots, x_n$ is expressed as:

$a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1$

$a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2$

$\vdots$

$a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n = b_m$

• $a_{ij} \in \mathbb{R}$ are the system coefficients, and $b_i \in \mathbb{R}$ are the constant terms.

  • Solution and Consistency Definitions:

• Solution: An ordered set of numbers $x_1, x_2, \dots, x_n$ satisfying all $m$ equations simultaneously.

• Consistent System: A system possessing at least one solution.

• Inconsistent System: A system possessing no valid solution.

  • Matrix Form Representation:

• System equation: $AX = B$, where:

$A = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{pmatrix}_{m \times n}, \quad X = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}_{n \times 1}, \quad B = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}_{m \times 1}$

• Augmented Matrix Representation:

  • Denoted $[A : B]$ and defined as:

$[A : B] = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} & : & b_1 \\ a_{21} & a_{22} & \dots & a_{2n} & : & b_2 \\ \vdots & \vdots & \ddots & \vdots & : & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} & : & b_m \end{pmatrix}$

Linear Homogeneous Systems

• System Definition:

  • When $b_1 = b_2 = \dots = b_m = 0$ ($B = 0$), the system $AX = 0$ is called a system of linear homogeneous equations.

  • Homogeneous systems always admit the zero or trivial solution $x_1 = x_2 = \dots = x_n = 0$ ($X = 0$).

  • Because the trivial solution always exists, a system of linear homogeneous equations is inherently consistent.

• Solution Procedure:

  1. Represent the system as an augmented matrix $[A : 0]$.

  2. Transform $[A : 0]$ into its corresponding row reduced echelon form to identify $r = \text{rank}(A)$.

  3. Evaluate system criteria against the number of unknowns $n$:

  • Case 1 ($r = n$): System possesses only the trivial solution ($x_1 = x_2 = \dots = x_n = 0$).

  • Case 2 ($r < n$): System possesses infinitely many non-trivial solutions. Assign arbitrary parameters to $n - r$ free variables and solve for the remaining variables using echelon back-substitution.

• Problems and Applied Scenarios:

  • Problem 1: Find the solution to the system:

$x + y - z = 0$

$2x + 4y - z = 0$

$3x + 2y + 2z = 0$

• Problem 2 (Distributed Database System): A distributed database consists of three servers whose synchronization errors satisfy:

$x + y - z = 0$

$2x - 3y + z = 0$

$x - 4y + 2z = 0$

* Task: Determine the complete solution set and interpret the network synchronization error state.

• Problem 3: Solve the system:

$2x - y - 3z = 0$

$-x + 2y - 3z = 0$

$x + y + 4z = 0$

• Problem 4 (Communication Network Flow Balancing): A computer network consists of three interconnected channels with data flow rate adjustments $x_1, x_2, x_3$ satisfying:

$3x_1 + 5x_2 - 4x_3 = 0$

$-3x_1 - 2x_2 + 4x_3 = 0$

$6x_1 + x_2 - 8x_3 = 0$

* Task: Determine the complete solution and evaluate whether data flow adjustments are independent or dependent.

• Problem 5: Solve the system:

$2x + 2y + 4z = 0$

$w - y - 3z = 0$

$2w + 3x + y + z = 0$

$-2w + x + 3y - 2z = 0$

Linear Non-Homogeneous Systems and Gaussian Elimination

• System Definition:

  • A linear system $AX = B$ where at least one $b_i \neq 0$ ($B \neq 0$) is called a non-homogeneous system.

• Gaussian Elimination Procedure:

  1. Construct the augmented matrix $[A : B]$.

  2. Perform elementary row operations to reduce $[A : B]$ to row reduced echelon form.

  3. Determine $\text{rank}(A)$ and $\text{rank}([A : B])$.

  4. Apply Consistency Rules:

  • Rule 1 ($\text{rank}(A) = \text{rank}([A : B]) = n$): The system is consistent and yields a unique solution.

  • Rule 2 ($\text{rank}(A) = \text{rank}([A : B]) = r < n$): The system is consistent and yields infinitely many solutions (parameterize $n - r$ free variables).

  • Rule 3 ($\text{rank}(A) \neq \text{rank}([A : B])$): The system is inconsistent and possesses no solution.

• Problems and Applied Scenarios:

  • Problem 1: Verify consistency and solve if solutions exist:

$x + y + 2z = 9$

$2x + 4y - 3z = 1$

$3x + 6y - 5z = 0$

• Problem 2 (Cloud Computing Resource Allocation): A cloud system allocates CPU capacity ($x_1$), memory capacity ($x_2$), and storage capacity ($x_3$) under requirements:

$2x_1 - x_2 + x_3 = 3$

$4x_1 + x_2 + 3x_3 = 15$

$x_1 - x_2 + 5x_3 = 14$

* Task: Solve the system and interpret the resource allocation state.

• Problem 3: Verify consistency and solve if solutions exist:

$3x_1 + 2x_2 + x_3 = 3$

$2x_1 + x_2 + x_3 = 0$

$6x_1 + 2x_2 + 4x_3 = 6$

• Problem 4: Verify consistency and solve if solutions exist:

$x + 2y + z = 3$

$2x + 5y - z = -4$

$3x - 2y - z = 5$

• Problem 5 (Parallel Computing Load Balancing): Processor workloads $x_1, x_2, x_3$ across three computing units satisfy:

$5x_1 - x_2 + 2x_3 = 7$

$-2x_1 + 6x_2 + 9x_3 = 0$

$7x_1 - 5x_2 - 3x_3 = 7$

* Task: Solve the system and interpret the computational load allocation.

LU Decomposition Method

• Definition and Existence Condition:

  • The factorization of a square matrix $A$ into $A = LU$, where $L$ is a unit lower triangular matrix and $U$ is an upper triangular matrix, is called LU decomposition.

  • Decomposition requires all principal minors of $A$ to be non-singular ($\det \neq 0$):

$a_{11} \neq 0, \quad \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} \neq 0, \quad \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \neq 0$

• Mathematical Formulation for $3 \times 3$ Systems:

  • Given system $AX = B$, set $A = LU$:

$L = \begin{pmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{pmatrix}, \quad U = \begin{pmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{pmatrix}$

• Step 1: Write $LUX = B$. Set $UX = V$ where $V = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$.

• Step 2: Solve $LV = B$ for $V$ using forward substitution:

$v_1 = b_1$

$l_{21}v_1 + v_2 = b_2$

$l_{31}v_1 + l_{32}v_2 + v_3 = b_3$

• Step 3: Solve $UX = V$ for $X$ using back substitution:

$u_{11}x_1 + u_{12}x_2 + u_{13}x_3 = v_1$

$u_{22}x_2 + u_{23}x_3 = v_2$

$u_{33}x_3 = v_3$

• Matrix Entry Determination Formulas:

  • Equating entries in $LU = A$ yields:

$u_{11} = a_{11}, \quad u_{12} = a_{12}, \quad u_{13} = a_{13}$

$l_{21}u_{11} = a_{21}, \quad l_{21}u_{12} + u_{22} = a_{22}, \quad l_{21}u_{13} + u_{23} = a_{23}$

$l_{31}u_{11} = a_{31}, \quad l_{31}u_{12} + l_{32}u_{22} = a_{32}, \quad l_{31}u_{13} + l_{32}u_{23} + u_{33} = a_{33}$

• Problems and Applied Scenarios:

  • Problem 1: Solve using LU decomposition:

$2x + 6y + 2z = 2$

$3x + 8y = -2$

$4x + 9y + 2z = 3$

• Problem 2 (Traffic Signal Optimization): Intersection green light durations $x, y, z$ (in minutes) satisfy:

$x + 2y + 3z = 14$

$2x + 3y + 4z = 20$

$3x + 4y + z = 14$

* Task: Solve using LU decomposition and determine individual green signal durations.

• Problem 3: Solve using LU decomposition:

$2x - y + 3z = 9$

$x + y + z = 6$

$x - y + z = 2$

• Problem 4 (Cybersecurity Risk Assessment): Risk factor weights for vulnerability severity ($x_1$), threat frequency ($x_2$), and data exposure impact ($x_3$) satisfy:

$3x_1 + 6x_2 - 3x_3 = 3$

$6x_1 + 15x_2 - 5x_3 = 11$

$-x_1 - 2x_2 + 6x_3 = 9$

* Task: Solve using LU decomposition and interpret the resulting factor weights.

Eigenvalues and Eigenvectors

• Definitions:

  • For an $n \times n$ matrix $A$, a non-zero vector $X \in \mathbb{R}^n$ is an eigenvector of $A$ if:

$AX = \lambda X \quad \implies \quad (A - \lambda I)X = 0$

• The scalar $\lambda$ is an eigenvalue of $A$, and $X$ is the eigenvector corresponding to $\lambda$

  • Characteristic Equation:

• A scalar $\lambda$ is an eigenvalue of $A$ if and only if it satisfies:

$\det(A - \lambda I) = 0 \quad \text{or} \quad |A - \lambda I| = 0$

• For a $3 \times 3$ matrix:

$\lambda^3 - s_1 \lambda^2 + s_2 \lambda - s_3 = 0$

    where $s_1 = \text{trace}(A) = \text{sum of leading diagonal elements}$, $s_2 = \text{sum of principal minors of } A$, and $s_3 = \det(A)$.

• For a $2 \times 2$ matrix:

$\lambda^2 - s_1 \lambda + s_2 = 0$

    where $s_1 = \text{trace}(A) = \text{sum of leading diagonal elements}$ and $s_2 = \det(A)$.

• Problems and Applied Scenarios:

  • Problem 1: Find all eigenvalues and corresponding eigenvectors of:

$\begin{pmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{pmatrix}$

• Problem 2 (Network Adjacency Analysis): Three network nodes have adjacency matrix:

$A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}$

* Task: Determine eigenvalues and eigenvectors, and interpret their significance regarding network connectivity.

• Problem 3: Find all eigenvalues and corresponding eigenvectors of:

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

• Problem 4 (Data Cluster Similarity): Similarity relationships among three clusters in data mining are given by:

$A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$

* Task: Determine eigenvalues and eigenvectors, interpreting their context in clustering.

• Problem 5: Find all eigenvalues and corresponding eigenvectors of:

$\begin{pmatrix} 2 & -2 & 2 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{pmatrix}$

• Problem 6: Find all eigenvalues and corresponding eigenvectors of:

$\begin{pmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{pmatrix}$

• Problem 7: Find all eigenvalues and corresponding eigenvectors of:

$\begin{pmatrix} -1 & 3 \\ 2 & 0 \end{pmatrix}$

Properties of Eigenvalues

• Fundamental Mathematical Properties:

  • Property a: A square matrix $A$ and its transpose $A^T$ share identical eigenvalues.

  • Property b: The sum of the eigenvalues of $A$ equals the trace of $A$ ($\sum \lambda_i = \text{trace}(A)$).

  • Property c: The product of the eigenvalues of $A$ equals $\det(A)$ ($\prod \lambda_i = \det(A)$).

  • Property d: Given eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ of matrix $A$:

  • (i) Matrix $kA$ ($k \neq 0$) has eigenvalues k\lambda_1, k\lambda_2, \dots, k\lambda_n$.\n * (ii) Matrix power A^p$($p > 0$) has eigenvalues$\lambda_1^p, \lambda_2^p, \dots, \lambda_n^p$.

  • (iii) Inverse matrix $A^{-1}$ has eigenvalues $\frac{1}{\lambda_1}, \frac{1}{\lambda_2}, \dots, \frac{1}{\lambda_n}$ provided \lambda_r \neq 0$.\n * **Property e:** Matrix A \pm kI$has eigenvalues$\lambda_1 \pm k, \lambda_2 \pm k, \dots, \lambda_n \pm k$.

  • Property f: Eigenvalues of a real symmetric matrix are strictly real numbers.

  • Property g: If zero ($0$) is an eigenvalue of matrix $A$, then $A$ is singular ($\det(A) = 0$).

  • Property h: For a triangular matrix or diagonal matrix, the eigenvalues are the leading diagonal entries.

  • Property i: Eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are mutually orthogonal.

• Analytical Problems on Properties:

  • Problem 1: Find the sum and product of eigenvalues of:

$\begin{pmatrix} 3 & 1 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 5 \end{pmatrix}$

• Problem 2: Two eigenvalues of $A = \begin{pmatrix} 4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -5 & -2 \end{pmatrix}$ are equal and double the magnitude of the third. Find the eigenvalues of A^2$.\n * **Problem 3:** If one eigenvalue of A = \begin{pmatrix} 1 & -3 & 3 \ 3 & -5 & 3 \ 6 & -6 & 4 \end{pmatrix}$is$4$, and the product of the remaining two eigenvalues is$4$, find all eigenvalues of$A$.

• Problem 4: Given a $3 \times 3$ matrix $A$ with one eigenvalue equal to $2$, sum of all eigenvalues equal to $4$, and product of all eigenvalues equal to $-26$, determine all three eigenvalues.

• Problem 5: Given eigenvalues $2, -1, -2$ for matrix $A$, find the eigenvalues of A^2 - 2I$.\n * **Problem 6:** Find the eigenvalues of A^2$and$A^{-1} for matrix:\n\nA = \begin{pmatrix} 3 & 0 & 0 \ 5 & 4 & 0 \ 3 & 6 & 1 \end{pmatrix}\n\n# Diagonalization of Matrices\n\n* **Similarity of Matrices:**\n * Matrix B$is similar to$A$if there exists an invertible matrix$P$such that$B = P^{-1}AP$.

• If $B$ is similar to $A$, then $A$ is similar to $B$ via $A = Q^{-1}BQ$, where $Q = P^{-1}$.

  • Diagonalization Definition:

• A square matrix $A$ is diagonalizable if there exists an invertible matrix $P$ such that $P^{-1}AP = D$ is a diagonal matrix. The matrix $P$ diagonalizes $A$.

  • Diagonalization Procedure:

• Find all eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ and corresponding linear independent eigenvectors $X_1, X_2, \dots, X_n$ of matrix A$.\n 2. Construct modal matrix P = \begin{pmatrix} X_1 & X_2 & \dots & X_n \end{pmatrix}_{n \times n}.\n 3. Compute inverse matrix P^{-1}.\n 4. Apply transformation P^{-1}AP = D$, where$D is the resulting diagonal matrix containing eigenvalues along its main diagonal.\n\n* **Problems and Applied Scenarios:**\n * **Problem 1:** Transform matrix to diagonal form:\n\n\begin{pmatrix} 1 & 1 & 3 \ 1 & 5 & 1 \ 3 & 1 & 1 \end{pmatrix}\n\n * **Problem 2 (Communication Network Modes):** Diagonalize adjacency matrix A = \begin{pmatrix} 0 & 1 & 1 \ 1 & 0 & 1 \ 1 & 1 & 0 \end{pmatrix} to determine independent communication modes.\n * **Problem 3:** Express matrix in diagonal form:\n\n\begin{pmatrix} 1 & 2 \ 2 & 4 \end{pmatrix}\n\n * **Problem 4:** Express matrix in diagonal form:\n\n\begin{pmatrix} 1 & 4 \ 2 & 3 \end{pmatrix}\n\n * **Problem 5 (Traffic Flow Modes):** Diagonalize road junction interaction matrix:\n\n\begin{pmatrix} 8 & -6 & 2 \ -6 & 7 & -4 \ 2 & -4 & 3 \end{pmatrix}\n\n * Task: Use diagonal form to identify independent traffic-flow modes.\n\n * **Problem 6:** Express matrix in diagonal form:\n\n\begin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix}\n\n * **Problem 7:** Express matrix in diagonal form:\n\n\begin{pmatrix} 7 & 2 \ -4 & 1 \end{pmatrix}\n\n * **Problem 8:** Express matrix in diagonal form:\n\n\begin{pmatrix} 2 & 3 \ 4 & 1 \end{pmatrix}\n\n# Quadratic Forms\n\n* **Mathematical Definition:**\n * A homogeneous expression of the second degree in any number of variables is called a quadratic form.\n * Examples: 2x^2 + 5xy - 7y^2$,$3x_1^2 - 4x_1x_2 + 7x_2^2$.

  • General Algebraic Form:

• For variable vector $X = \begin{pmatrix} x_1 & x_2 & \dots & x_n \end{pmatrix}^T$:

$Q = \sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j \quad (a_{ij} = a_{ji})$

$Q = a_{11}x_1^2 + a_{22}x_2^2 + \dots + a_{nn}x_n^2 + 2a_{12}x_1x_2 + 2a_{13}x_1x_3 + \dots + 2a_{(n-1)n}x_{n-1}x_n$

$Q = \sum_{i=1}^n a_{ii} x_i^2 + 2 \sum_{i=1}^{n-1} \sum_{j=i+1}^n a_{ij} x_i x_j$

• Matrix Representation:

  • System expression: $Q = X^T A X$, where $A$ is a real symmetric matrix ($A = A^T$) of order $n \times n$, called the matrix of the quadratic form.

  • Example: $Q = 2x^2 + 6xy - 5y^2 = \begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 3 & -5 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$.

• Problems:

  • Problem 1: Find matrix of quadratic form 2x_1^2 - 2x_2^2 + 4x_3^2 + 2x_1x_2 - 6x_1x_3 + 6x_2x_3$.\n * **Problem 2:** Find matrix of quadratic form 10x_1^2 + 2x_2^2 + 5x_3^2 - 4x_1x_2 + 6x_2x_3 - 10x_1x_3$.

  • Problem 3: Find quadratic form corresponding to matrix:

$\begin{pmatrix} 0 & -1 & 2 \\ -1 & 1 & 4 \\ 2 & 4 & 3 \end{pmatrix}$

• Problem 4: Find quadratic form corresponding to matrix:

$\begin{pmatrix} 1 & 2 & 5 \\ 2 & 4 & 6 \\ 5 & 6 & 3 \end{pmatrix}$

Orthogonality and Vector Normalization

• Orthogonal Matrix Definition:

  • A square matrix $A$ is orthogonal if $A^{-1} = A^T$, which implies A A^T = A^T A = I$.\n\n* **Pairwise Orthogonality:**\n * Vectors X_1, X_2, X_3 are pairwise orthogonal if:\n\nX_1 \cdot X_2^T = X_2 \cdot X_3^T = X_3 \cdot X_1^T = 0\n\n * Note: A matrix whose columns are pairwise orthogonal and normalized is an orthogonal matrix.\n\n* **Vector Normalization:**\n * The normalized vector of X = \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} is given by:\n\nX^N = \begin{pmatrix} \frac{x_1}{d} \ \frac{x_2}{d} \ \frac{x_3}{d} \end{pmatrix} \quad \text{where } d = \sqrt{x_1^2 + x_2^2 + x_3^2}\n\n# Canonical Form of Quadratic Forms and Nature Classification\n\n* **Canonical Form Transformation:**\n * For quadratic form Q = X^T A X$, transformation$X = PY$(where$P$is the normalized modal matrix formed by normalized orthogonal eigenvectors of$A$) reduces$Q to:\n\nQ = X^T A X = (PY)^T A (PY) = Y^T (P^T A P) Y = Y^T D Y\n\nQ = \lambda_1 y_1^2 + \lambda_2 y_2^2 + \dots + \lambda_n y_n^2\n\n * where D = \begin{pmatrix} \lambda_1 & 0 & \dots & 0 \ 0 & \lambda_2 & \dots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \dots & \lambda_n \end{pmatrix}.\n\n\n\n* **Nature Classification of Quadratic Form:**\n * A real quadratic form Q = X^T A X$in$n variables is classified as:\n * **Positive Definite:** All eigenvalues of A$are strictly positive ($\lambda_i > 0).\n * **Negative Definite:** All eigenvalues of A$are strictly negative ($\lambda_i < 0).\n * **Positive Semidefinite:** All eigenvalues of A$are non-negative ($\lambda_i \ge 0) with at least one eigenvalue equal to zero.\n * **Negative Semidefinite:** All eigenvalues of A$are non-positive ($\lambda_i \le 0) with at least one eigenvalue equal to zero.\n * **Indefinite:** Eigenvalues of A consist of both positive and negative values.\n\n* **Reduction Procedure Step-by-Step:**\n 1. Obtain symmetric matrix A$from quadratic form$Q = X^T A X.\n 2. Compute all eigenvalues \lambda_1, \lambda_2, \dots, \lambda_n$and corresponding eigenvectors$X_1, X_2, \dots, X_n.\n 3. Verify pairwise orthogonality of eigenvectors (orthogonalize if degenerate) and compute normalized eigenvectors X_1^N, X_2^N, \dots, X_n^N.\n 4. Form normalized modal matrix P = \begin{pmatrix} X_1^N & X_2^N & \dots & X_n^N \end{pmatrix}.\n 5. Apply substitution X = PY$to express canonical form$Q = \lambda_1 y_1^2 + \lambda_2 y_2^2 + \dots + \lambda_n y_n^2$.

  1. Determine nature classification based on eigenvalue signs.

• Problems and Applied Scenarios:

  • Problem 1 (Traffic Congestion Model): Congestion across three junctions is modeled by $x^2 + 5y^2 + z^2 + 2xy + 2yz + 6xz$.

  • Task: Find canonical form and determine nature of traffic network.

  • Problem 2: Find canonical form and determine nature of quadratic form 2x_1x_2 + 2x_1x_3 + 2x_2x_3$.\n * **Problem 3:** Find canonical form and determine nature of quadratic form 2x^2 + 2y^2 + 2z^2 + 2xy + 2yz + 2xz$.

  • Problem 4 (Network Router Performance): Performance across three routers is measured by 8x_1^2 + 7x_2^2 + 3x_3^2 - 12x_1x_2 + 4x_1x_3 - 8x_2x_3$.\n * Task: Determine canonical form and classify nature of network performance model.\n * **Problem 5:** Find canonical form and determine nature of quadratic form -2x^2 + 4xy - 2y^2$.

  • Problem 6: Find canonical form and determine nature of quadratic form $$x^2 - 4xy + y^2$.