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Linear Algebra – Eigenvalues and Eigenvectors

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    Study Guide Linear Algebra – Eigenvalues and Eigenvectors 1. Determining the Eigenvalues of a Matrix 1. Why Do We Care About Eigenvalues? In linear algebra, every linear operator can be represented by multiplying by a square matrix. So instead of thinking about abstract operators, we can just study square matrices . That means: Finding the eigenvalues and eigenvectors of a linear operator is the same as finding them for a square matrix. Throughout this chapter, we assume all matrices are square matrices . 2. What Is an Eigenvalue? Let’s start with a square matrix (A). A number ( λ ) (lambda) is called an eigenvalue of (A) if there exists a nonzero vector (x) such that: This means: • When matrix (A) acts on vector (x), • It only stretches or shrinks the vector, • It does not change its direction. That special vector (x) is called an eigenvector .

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    Study Guide 3. Turning the Equation Into a Useful Form We start with: Now rewrite it step by step: Factor out (x): Here: • (I) is the identity matrix. • (A − λI ) means subtract ( λ ) from every diagonal entry of (A). 4. When Does a Nonzero Solution Exist? We want a nonzero vector (x). For the system to have a nonzero solution, the determinant must be zero: This equation is called the characteristic equation . The expression:

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    Study Guide is called the characteristic polynomial . Important facts: • It is a polynomial in ( λ ). • It has degree (n) if (A) is an ( n × n ) matrix. • The roots of this polynomial are the eigenvalues . Example 1: Finding Eigenvalues of a 2×2 Matrix We are given: Step 1: Form (A − λ I ) Subtract ( λ ) from each diagonal entry: Step 2: Take the Determinant Now simplify: This is the characteristic polynomial . Step 3: Solve the Characteristic Equation

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    Study Guide Set it equal to zero: Factor: So the eigenvalues are: A Small Note About Notation Sometimes books write: instead of For: • Even - sized matrices → they are exactly the same. • Odd - sized matrices → they differ by a negative sign. But this does not affect the eigenvalues , because both expressions become zero at the same values of ( λ ). Example 2: 3 × 3 Checkerboard Matrix We are given:

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    Study Guide Step 1 : Form (C − λ I ) Subtract ( λ ) from the diagonal: Step 2: Compute the Determinant To simplify the determinant: • Add the second row to the third row. • Then expand along the first column (Laplace expansion). After simplifying carefully, we get: Step 3: Solve the Characteristic Equation Set equal to zero: This gives:

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    Study Guide So the eigenvalues are: • ( λ ) (with multiplicity 2) • ( λ = 3) 2. Determining the Eigenvectors of a Matrix 1. Big Idea: How Do We Find Eigenvectors? Before finding eigenvectors, we must first know the eigenvalues . Why? Because eigenvectors are found by plugging each eigenvalue into the equation: This can also be written as: Now we solve this system for x . Important: • We are looking for nonzero solutions . • Those nonzero solutions form the eigenvectors corresponding to that eigenvalue . • We repeat the process for each eigenvalue. Example 1: Finding Eigenvectors of a 2×2 Matrix We use the matrix:

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    Study Guide From the previous section, we already found the eigenvalues: Now let’s find the eigenvectors for each one. Eigenvectors for λ = −1 We substitute λ = −1 into: So we solve: This gives a system of equations. After simplifying, we obtain: Notice something important: These two equations are not independent . They are actually the same equation written twice. That means we have infinitely many solutions. From: So every solution looks like:

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    Study Guide which can be written as: where t is any nonzero scalar . Therefore, the eigenvectors corresponding to λ = −1 are all multiples of: Eigenvectors for λ = −2 Now substitute λ = −2 into: After simplifying, we get the system: Again, the equations are not independent. Solving: This gives vectors of the form:

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    Study Guide where t ≠ 0 . Therefore, the eigenvectors corresponding to λ = −2 are all multiples of: Understanding What Happened For each eigenvalue: 1. Substitute λ into (A − λI ) ( x = 0) 2. Solve the resulting system. 3. If calculations are correct, there must be nonzero solutions . 4. The solutions form a line (or subspace) of eigenvectors. If you ever find that only the zero vector works, it means something went wrong — because eigenvalues must produce nonzero eigenvectors. Example 2: The General 2×2 Matrix Now consider a general matrix: Let’s explore what we can learn about its eigenvalues. Step 1: Find the Characteristic Equation We compute:

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    Study Guide This gives: This is a quadratic equation in λ. Using the quadratic formula: Step 2: Understanding the Discriminant The expression under the square root is: This can be rewritten as: Now suppose the matrix is symmetric , meaning: Then the expression becomes: This is the sum of two squares. And a sum of squares is always nonnegative . Therefore: If a matrix is real and symmetric, its eigenvalues are always real numbers.

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