3: Determinants and Diagonalization
- Page ID
- 58847
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The cofactor expansion is stated (proved by induction later) and used to define determinants inductively and to deduce the basic rules. The product and adjugate theorems are proved. Then the diagonalization algorithm is presented (motivated by an example about the possible extinction of a species of birds). As requested by our Engineering Faculty, this is done earlier than in most texts because it requires only determinants and matrix inverses, avoiding any need for subspaces, independence and dimension. Eigenvectors of a 2×2 matrix A are described geometrically (using the A-invariance of lines through the origin). Diagonalization is then used to study discrete linear dynamical systems and to discuss applications to linear recurrences and systems of differential equations. A brief discussion of Google PageRank is included.
- 3.1: The Cofactor Expansion
- This page covers the determinant of square matrices, starting with definitions and methods for \(2 \times 2\) and \(3 \times 3\) matrices, extending to larger ones using cofactor expansion. It discusses consistent calculation across rows and columns, along with properties of determinants affected by row operations. The text includes methods for special matrices like the Vandermonde matrix and triangular matrices, with emphasis on how determinants behave as linear transformations.
- 3.2: Determinants and Matrix Inverses
- This page covers key concepts related to determinants and their properties, emphasizing that a square matrix is invertible if \(\det A \neq 0\). It introduces the Product Theorem, adjugates, and cofactors, illustrating how they relate to calculating inverses and Cramer's Rule for solving linear systems. The significance of the Vandermonde determinant in polynomial interpolation is highlighted, along with methods for determining polynomials that fit given data points.
- 3.3: Diagonalization and Eigenvalues
- This textbook covers systems evolving over time, focusing on populations and their modeling through diagonalization in linear algebra. Key concepts include eigenvalues, eigenvectors, characteristic polynomials, and diagonalization processes. The relationship between matrices and their dynamics is examined, including the Cayley-Hamilton theorem and applications in ecological models.
- 3.4: An Application to Linear Recurrences
- This page explores recursive sequences through combinatorial problems, emphasizing linear recurrence relations like \(x_{k+2} = x_{k+1} + x_k\). It introduces methods such as matrix diagonalization to derive explicit formulas for sequences, connecting it to the Fibonacci sequence. The page highlights the Fibonacci sequence's significance in nature and introduces the Binet formula.
- 3.5: An Application to Systems of Differential Equations
- This page covers the concept of differentiable functions and their derivatives, specifically focusing on systems of first-order differential equations. It introduces solving methods through diagonalization, emphasizing matrix representation with eigenvalues and eigenvectors.
- 3.6: Proof of the Cofactor Expansion Theorem
- This page provides an inductive definition of the determinant for \(n \times n\) matrices, starting with \(1 \times 1\) matrices, and explains cofactor expansion. It includes methods for computing determinants, illustrates properties under row operations, and outlines key theorems regarding the effects of row interchanges and identical rows on the determinant.
- 3.E: Supplementary Exercises for Chapter 3
- This page covers properties and exercises of determinants and matrix algebra, including demonstrating determinant equivalence between a matrix and its transpose. It explores calculations for specific matrix forms, manipulations through row operations, and relationships among determinants. Additionally, it includes examples that emphasize these concepts and touches on eigenvalues in matrix theory.
Thumbnail: The volume of this parallelepiped is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3. (CC BY 3.0; Claudio Rocchini via Wikipedia)