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Chapter 4 - Eigenvalues and Eigenvectors

# Chapter 4 - Eigenvalues and Eigenvectors - Chapter 4...

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Chapter 4 – Eigenvalues and Eigenvectors Section 4.1 – Introduction to Eigenvalues and Eigenvectors Pg 253, Definition – Let A be an matrix. A scalar λ is called an eigenvalue of A if there is a nonzero vector x such that . Such a vector x is called an eigenvector of A corresponding to λ . Pg 255, Definition – Let A be an matrix and let λ be an eigenvalue of A. The collection of all eigenvectors corresponding to λ , together with the zero vector, is called the eigenspace of λ and is denoted by E λ . Section 4.2 – Determinents Pg 263, Definition – Let . Then the determinant of A is the scalar Pg 264, Definition – Let be an matrix, where . Then the determinant of A is the scalar Pg 265, Theorem 4.1 – The Laplace Expansion Theorem – The determinant of an matrix , where can be computed as (which is the cofactor expansion along the ith row ) and also as (the cofactor expansion along the jth column ). Pg 268, Theorem 4.2 – The determinant of a triangular matrix is the product of the entries on its main diagonal. Specifically, if is an triangular matrix then Pg 268, Theorem 4.3 – Properties of Determinants – Let be a square matrix. a. If A has a zero row (column), then . b. If B is obtained by interchanging two rows (columns) of A, then . c. If A has two identical rows (columns), . d. If B is obtained by multiplying a row (column) of A by k, then . e. If A, B, and C are identical except that the ith row (column) of C is the sum of the ith rows (columns) of A and B, then .

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