Rishit Matrices Notes PDF

# Module - 3 ## Eigen Values & Eigen Vectors **Given a square matrix Amxn** - **Characteristic polynomial:** - f(x) = det (A-λI) - **Characteristic equation:** - det (A-λI)=0 - **The n roots of the characteristic equation are called eigen values:** - λ = λ₁, λ₂, λ₃, .... λₙ **Example** -...

# Module - 3 ## Eigen Values & Eigen Vectors **Given a square matrix Amxn** - **Characteristic polynomial:** - f(x) = det (A-λI) - **Characteristic equation:** - det (A-λI)=0 - **The n roots of the characteristic equation are called eigen values:** - λ = λ₁, λ₂, λ₃, .... λₙ **Example** - Let the matrix be A = [1 2] [3 4] - A - λI = [1 2] - λ [1 0] [3 4] [0 1] - A - λI = [1-λ 2] [3 4-λ] - f(λ) = det (A - λI) - f(λ) = (1-λ) (4-λ) - 8 - f(λ) = 3 - 4λ + λ² - 8 - f(λ) = λ² -4λ - 5 (Characteristic Polynomial) - char eq: λ² - 4λ - 5 = 0 - (λ - 5) (λ + 1) = 0 - λ = 5, -1 (Eigen values) **Example** B = 5A⁴ - 7A³ + 2A - 3A¹ - Eigen values of B = 5(1)⁴ - 7(1)³ + 2(1) - 3(1)¹ - 1 = 5(2)⁴ - 7(2)³ + 2(2) - 3(2)¹ - 1 = 5(3)⁴ - 7(3)³ + 2(3) - 3(3)¹ - 1 = -3, 26.5, 221 **Example** A = [-2 2 -3] [2 -1 -6] [-1 -2 6] - λ³ - (-1)² + (-12 - 3 - 6)λ - 45 = 0 - λ³ + λ² - 21λ - 45 = 0 - λ = 5, -3, 7/3 - By prop 1 - 5 + (-3) + 7/3 = -1 - 7/3 = -3 - Eigenvalues of A are 5, -3, -3 - B=A³ - 5A² + 3A - SI - Eigenvalues of B = 5³ - 5(5)² + 3(5) - 5 = (-3)³ - 5(-3)² + 3(-3) - 5 = (-3)³ - 5(-3)² + 3(-3) - 5 = -101 - 86 - 86 **Example** A= [2 0] [2 2] - A - λI = [2 0] - λ[1 0] [2 2] [0 1] - A - λI = [2 - λ 0] [2 2 - λ] - f(λ) = det (A - λI) - f(λ) = (2 - λ) (2 - λ) - 0 - f(λ) = 4 - ... A is a lower triangular matrix - λ = 2, 2 - det A = 4 - eigenvalue of Adj A = 1/4, 1/4 - = 2/2, 2/2 - = 2, 2 **Example** A=[ 1 0 -1 ] [ 1 2 2 ] [2 2 3] - 2³ - 6λ² + { 2 * 2 * 3 + 1 * 1 * 1 + 1 * 1 * 2 } λ - det A = 0 - 2³ - 6λ² + 11λ - 6 = 0 - λ = 1, 2, 3 **Properties** - **Sum of eigenvalues = trace A** - **Product of eigenvalues = det A** - **If A is a 2x2 matrix** - char eq is λ² - (trace A)λ + det A = 0 - **If A is a 3x3 matrix** - char eq is λ³ - (trace A)λ² +(c₁₁ + c₂₂ + c₃₃)λ - det A = 0 - **If A is a diagonal matrix, an upper triangular matrix, or a lower triangular matrix, then the diagonal elements are the eigenvalues of A.** - **If λ is an eigenvalue of A, then g(λ) is an eigenvalue of g(A)** - **Eigen values of Adj A are det (A)/λ** **Example** - A = [ 1 0 -1 ] [ 1 2 2] [2 2 3] - λ = 1, 2, 3 - λ = 1 - (A - λI)x = 0 - [ 0 0 -1 ] x₁ = 0 - [ 1 1 2 ] x₂ = 0 - [ 2 2 2 ] x₃ = 0 - By Cramer's Rule, - x₁ = x₂ = - x₃ - x = t (1, -1, 0) is eigenvector for λ = 1 - λ = 2 - (A - λI)x = 0 - [ -1 0 -1 ] x₁ = 0 - [ 1 0 2 ] x₂ = 0 - [ 2 2 1 ] x₃ = 0 - By Cramer's Rule, - x₁ = - 2x₃ - x = t (-2, 1, 1) is eigenvector for λ = 2 - λ = -1 - x = t [-2] is eigenvector of A for λ=-1 - A = [ 2 0 ] [ 2 2 ] - λ = 2, 2 - for λ = 2, - [ 0 0 ] = 0 - [ 2 0 ] x1 = 0 - x = t [ 0 ] t ≠0 is an eigenvector for λ = 2 **Eigenvectors of A:** - A non-zero row column vector X is called the eigenvector of A corresponding to eigenvalue λ. **Example** - A = [ 1 4] [ 2 -1] - λ = 5, -1 - for λ = 5 - (A - λI)x = 0 - [ -4 4 ] x₁ = 0 - [ 2 -6 ] x₂ = 0 - -4x₁ + 4x₂ = 0 & 2x₁ - 6x₂ = 0 - x₁ = x₂ - x = t [1] t ≠ 0 is an eigenvector of A for λ = 5 - for λ = -1 - (A - λI)x = 0 - [ 2 4 ] x₁ = 0 - [ 2 0 ] x₂ = 0 - 2x₁ + 4x₂ = 0 & 2x₁ + 4x₂ = 0 - - By Cramer's Rule, - x₁ = - x₂ = - x₃ - x = t (-1, -2, 1) is eigenvector for λ = -3 **Example** - A = [ 1 0 -1] [ 1 2 2] [ 2 2 3] - λ = 1, 2, 3 - for λ = 1 - (A - λI)x = 0 - [ 0 0 -1 ] x₁ = 0 - [ 1 1 2 ] x₂ = 0 - [ 2 2 2 ] x₃ = 0 - By Cramer's Rule, - x₁ = - x₂ = -x₃ - x = t (-1, -1, 2) is eigenvector for λ = 3 - A = [-2 2 -3] [ 2 -1 -6] [-1 -2 6] - λ = 5, -3, 7/3 - for λ = 5 - [ -7 2 -3 ] x₁ = 0 - [ 2 -6 -6] x₂ = 0 - [ -1 -2 -5] x₃ = 0 **Diagonalizable Matrices:** - Algebraic multiplicity of λ= no of times the eigen value appears as a root - Geometric multiplicity of λ= no of linearly independent eigenvectors found for eigen value λ - **If AM=GM λ of A then A is called Diagonalizable.** - **If A is diagonalizable, there exists diagonal matrix D & diagonalizing matrix P such that D=P⁻¹AP** - **NOTE:** A matrix with distinct eigen values is always diagonalizable **Example** - A = [ 1 4 -2 ] [ -2 4 -2 ] [ -2 4 2] - λ³ - 4λ² + 6λ - 6 = 0 - AM = 2, GM = 2 - D = [9 0 0] [ 0 18 0] [ 0 0 18] - P = [ -2 -1 1] [ 0 1 2] [ 1 2 0] - Since AM=GM λ of A - A is diagonalizable. **Example** - λ = 9, 18, 18 - for λ = 9 - (A - λI)x = 0 - [ 0 4 -2] x₁ = 0 - [ 4 5 2] x₂ = 0 - [ -2 2 8] x₃ = 0 - By Cramer's Rule - x₁ = - x₂ = - x₃ - x = t [-2] AM = 1, GM = 1 [ -1 ] [ 1 ] - for λ = 18 - (A - λI)x = 0 - [ -4 4 -2] x₁ = 0 - [ 4 -4 -2] x₂ = 0 - [ -2 2 8 ] x₃ = 0 **Example** - A = [ 1 0 -1] [ 1 2 2] [ 2 2 3] - λ = 1, 2, 3 - for λ = 1 - (A - λI)x = 0 - [ 0 0 -1 ] x₁ = 0 - [ 1 1 2 ] x₂ = 0 - [ 2 2 2 ] x₃ = 0 - D = [ 1 0 0 ] [ 0 2 0 ] [ 0 0 3 ] - P = [ 1 2 -1 ] [ -1 -1 1] [0 -2 2] - A = [13 5 -2] [ 3 7 2] [ 5 4 7] - λ³ - 27λ² + 9λ - 729 = 0 - λ = 9, 9, 9 - for λ = 9, - [ 4 5 -2] x₁ = 0 - [ 3 2 -8] x₂ = 0 - [ 5 4 -2 ] x₃ = 0 **Example** - A = [ 1 -2 0 ] [ 1 2 2] [ 1 2 3] - λ³ - 6λ² + (2 + 3 + 4)λ - 4 = 0 - λ³ - 6λ² + 9λ - 4 = 0 - λ = 4, 1, 1 - 4 + 1 + 1 = 6 - λ = 4, 1, 1 - for λ = 4 - (A - λI)x = 0 - [ -3 -2 0] x₁ = 0 - [ 1 -2 2] x₂ = 0 - [ 1 -2 -1] x₃ = 0 - By Cramer's Rule, - x₁ = -x₂ = - x₃ - x = t [ -2 ] AM = 1, GM = 1 [ 3 ] [ 4 ] - for λ = 1 - (A - λI)x = 0 - [ 0 -2 0] x₁ = 0 - [ 1 -1 2] x₂ = 0 - [ 1 -2 2 ] x₃ = 0 - By Cramer's Rule, - x₁ = - x₂ = -x₃ - x = t [-2] AM = 2, GM = 1 [ 0 ] [ 1 ] - ... A is not diagonalizable. **Example** - A = [1 3 7] [ 4 2 3] [ 1 1 2] - λ³ - 4λ² - 20λ - 35 = 0 - By Cayley Hamilton Theorem, - A³ - 4A² - 20A- 35I = 0 - f(A) = (A² - 4A² - 20A - 35I) (AHA) + (-A + 2A + I) - f(A) = - A² + 2A +I - f(A) = -20 [-23] +2 [ 1 3 7 ] + [1 0 0 ] [ 10 22 31 ] [ 4 2 3 ] [ 0 1 0 ] [ 19 9 14 ] [ 1 1 2 ] [ 0 0 1 ] - f(A) = [-17 -17 -4] [ -7 -17 31] [ -8 - 5 -11] - x₁ = - x₂ = - x₃ - x = t [- 2] AM = 3, GM = 1 [ 1] [ 2] - ... A is not diagonalizable. **Cayley Hamilton Theorem** - Every square matrix satisfies its own characteristic equation - A = [ 1 2 ] [ 2 3] - λ² - 4λ - 5 = 0 - By Cayley Hamilton Theorem, - A² - 4A - 5I = 0 **Example** - f(A) = A⁴ - 5A³ + 12A² + 3A -7I - A - 4A - SI = A⁴ - 5A³ + 12A² + 3A -7I - ...f(A) = 50 [ 1 0] + 58 [ 1 0] [ 2 1 ] [ 0 1 ] - ...f(A) = [ 108 208 ] [ 100 208 ] - By C-H, A² - 4A - SI = 0 - A¹(A² - 4A - SI) = A¹ x 0 - A¹ - 4A - 5A¹ = 0 - A¹ - 1 = (A - 4I) - A¹ = 1 (A - 4I) **Example** - A = [ -1 2 3 ] [ -1 1 0 ] [ 3 1 0 ] - λ³ - 6λ² + 10λ - 3 = 0 - By Cayley Hamilton Theorem, - A³ - 6A² + 10A - 3I = 0 - A³ = 6A² - 10A + 3I - A⁴ = 6(6A² - 10A + 3I) - 10A² + 3A - A⁴ = 26 A² - 57A + 118I - A² - 6A + 10I - 3A - I = 0 - A - I = 1 (A² - 6A + 10I) - A⁻¹ = 1/3 (A - 6I + 10A⁻¹) - = 1/3 (A - 6I + 10 (A² - 6A + 10I)) - A⁻¹ = 1/9 (10 A² - 19 A + 82 I) **Functions of a Square Matrix** - f(A) = pA + qI → 2x2 - f(A) = pA² + qA + rI → 3x3 **Example** - A = [ 4 7 ] [ 1 3 ] - λ² - 12λ + 11 = 0 - λ² - 11λ - λ + 11 = 0 - (λ - 11) (λ - 1) = 0 - λ = 1, 11 - A¹⁰⁰ = pA + qI - λ¹⁰⁰ = pλ + q - in λ = 11, 11¹⁰⁰ = p (11) + q - p + q = 1 - in λ = 11, 11¹⁰⁰ = p (11) + q - 11¹⁰⁰ = 11p + q - 11¹⁰⁰ - 1 = 10p - p = (11¹⁰⁰ - 1)/10 - p + q = 1 - q = 1 - (11¹⁰⁰ - 1)/10 - = - 11¹⁰⁰ + 11 / 10 - e¹ᵗ = p + q - = - pI + q - e¹ᵗ + e⁻¹ᵗ = 2q - = 2cos(t) - e¹ᵗ - e⁻¹ᵗ = 2pi - = 2sin(t) - eᵗᴬ = sin (t)A + cos (t)I - = sin (t) [0 0] + cos(t) [ 1 0 ] [ 1 0 ] [ 0 1 ] - = sin (t) [ 0 cos(t) ] [ sin (t) cos (t) ] - eᵃᵗ⁸ = [ 1 0] [ 0 -1] - eᵃᵗ⁸ = A - A¹⁰⁰ = (11¹⁰⁰ - 1)/10 [ 4 7 ] + (-11¹⁰⁰ + 11)/10 [ 1 0] [ 1 3 ] [ 0 1] - = ((11¹⁰⁰ - 1)/10 [ 4 7 ] [ 1 3] - = (11¹⁰⁰ - 1)/10 [ 4 7 ] + (11¹⁰⁰ - 1)/10 [ 1 0] [ 1 3 ] [ 0 1 ] - = (11¹⁰⁰ - 1)/10 [ 3 * 11 + 7 ] [ 10 ] [ 7 * 11 - 1] - = (11¹⁰⁰ - 1)/10 [ 3 * 11 + 7 ] [ 10 ] [ 7 * 11 - 1] - A = [ π/2 0 ] [ 0 3π/2 ] - λ² - 2πλ + 3π² = 0 - λ = π, 3π/2 - ... Upper triangular - sin A = pA + qI - sin λ = pλ + q - in λ = π, sin π = p(π) + q - 0 = pπ + q - in λ = 3π/2, sin 3π/2 = p (3π/2) + q - -1 = 3π/2 + q - 3π/2 + q = - 1 - P = 3 / π - 3π/2 + q = - 1 - q = - 1 - 3π/2 - sin A = -2A + 3πI - = -2 [ π/2 0 ] + 3π [ 1 0 ] [ 0 3π/2 ] [ 0 1 ] - = [ -π 0 ] + [ 3π 0 ] [ 0 - 3π ] [0 3π ] - = [ -2π 0] [ 0 -2π] - = [ 0 0 ] [ -2 -2π ] - = [ 0 0] [ 0 -2 ] - A = [ 0 1 ] [ -1 0 ] - λ² - 0λ + 1 = 0 - λ² +1 = 0 - λ = ±i - eᵗᴬ = pA + qI - e¹ᵗ = pλ + q - (e¹ᵗ - 1) (e⁻¹ᵗ - 1) = 0 - e¹ᵗ + e⁻¹ᵗ - e¹ᵗ - e⁻¹ᵗ + 1 = 0 - = 0 - e¹ᵗ + e⁻¹ᵗ - 1 = 0 - 2cos(t) -1 = 0 - cos(t) = 1/2 - t=± π/3 **Example** - A = [ 3 4 -1 ] [ 5 -2 1 ] [ 2 1 3 ] - λ³ - 6λ² + 9λ - 4 = 0 - λ³ - 6λ² + 9λ - 4 = 0 - λ = 2, 1, 1 - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A = [ 3 4 -1 ] [ 5 -2 1] [ 2 1 3] - λ³ - 6λ² + 9λ - 4 = 0 - λ³ - 6λ² + 9λ - 4 = 0 - λ = 2, 1, 1 - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ = 1/5 [ 2 4 -1 ] [ 1 3 -1 ] [ -1 -1 2 ] - A⁻¹ =