Matrix Diagonalization

University:
The University Of Western Ontario
Course:
CALC 1000A | Calculus I
Academic year:

2020
Views:

428

Pages:

3
Author:

Kyuwon J.

Tutorial 2 Problem 1: Matrix Diagonalization 6 2 Let A = . Determine if A is diagonalizable. If it is, find P and D such that 2 3 A = P DP −1 . Solution: 1. Find Eigenvalues: Solve det(A − λI) = 0: 6−λ 2 A − λI = 2 3−λ det(A − λI) = (6 − λ)(3 − λ) − (2)(2) = λ2 − 9λ + 14 = 0 Factorize: (λ − 7)(λ − 2) = 0 Eigenvalues: λ1 = 7, λ2 = 2. 2. Find Eigenvectors: For λ1 = 7: −1 2 −1 2 x1 A − 7I = , =0 2 −4 2 −4 x2 2 Eigenvector: v1 = . 1 For λ2 = 2: 4 2 A − 2I = 2 1 −1 Eigenvector: v2 = . 2 3. Construct P and D: 2 −1 7 0 P = ,D= 1 2 0 2 Thus, A = P DP −1 . — Problem 2: Rank and Nullity   1 2 3 Given the matrix B = 2 4 6: 1 3 4 1 1. Find the rank of B. 2. Determine the nullity of B. Solution: 1. Row Reduce to Row Echelon Form:     1 2 3 1 2 3 B = 2 4 6 → 0 0 0 1 3 4 0 1 1 The matrix has 2 non-zero rows, so rank(B) = 2. 2. Nullity: Using the Rank-Nullity Theorem: rank(B) + nullity(B) = number of columns 2 + nullity(B) = 3 =⇒ nullity(B) = 1 — Problem 3: Solve System Using LU Decomposition Solve the system: 2x + 4y − z = 7x + 3y + 2z = 133x − y + 4z = 10 using LU decomposition. Solution:   2 4 −1 2  as LU , where L is a lower 1. Decompose A = LU : Write A = 1 3 3 −1 4 triangular matrix and U is an upper triangular matrix. Using Gaussian elimination:     1 0 0 2 4 −1 L = 0.5 1 0 , U = 0 1 2.5 1.5 −2 1 0 0 8   7 2. Solve Ly = b: Substitute b = 13: 10      1 0 0 y1 7 0.5 1 0 y2  = 13 1.5 −2 1 y3 10 2 Solve step-by-step to find:   7 y = 9.5 −7 3. Solve U x = y:      2 4 −1 x1 7 0 1 2.5 x2  = 9.5 0 0 8 x3 −7 Solve step-by-step to find:   1 x= 2  −1 — Concepts Used and Their Importance 1. Diagonalization: Useful for simplifying powers of matrices and solving differential equations involving matrices. 2. Rank and Nullity: Crucial for determining the consistency and solvability of linear systems and understanding vector spaces. 3. LU Decomposition: Efficient for solving large linear systems and often used in numerical methods for stability and speed. These concepts are chosen to demonstrate a wide range of matrix applications in exams, such as linear algebra, applied mathematics, and computational techniques. 3