math-2011-note12

math-2011-note12 - Definition 36(p.161 An m × m matrix A =...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Definition 36 (p.161) . An m × m matrix A = ( a ij ) is called an upper-triangular matrix if a ij = for i > j. A is called a lower-triangular matrix if a ij = for i < j. A is called a diagonal matrix if a ij = for i negationslash = j. Theorem 56 (Fact 26.11, p.731) . The determinant of a triangularmatrix is the product of its diagonal entries • Denote C ij = (- 1 ) i + j det A ij . Then, det A = ∑ m j = 1 a ij C ij . • And ∑ m j = 1 a k j C ij = 0 if k negationslash = i , since ∑ m j = 1 a k j C ij = det A prime , where A prime is the matrix A with i-th row replaced by k-th row. Theorem 57 (Thm 26.7, p.736) . For a non-singular m × m matrix A, the inverse is A- 1 = 1 det A    C 11 ··· C m 1 . . . . . . . . . C 1 m ··· C mm    Theorem 58 (Thm 26.8, Cramer’s Rule, p.737) . Let A be a non-singularm × m matrix. The unique solution x = ( x 1 , ··· , x m ) T of the m linear equation system A x = b is x j = det B j det A , j = 1 , ··· , m , where B j is the matrix A with the j-th column of A replaced by b . 16.4 Diagonalization by Completion of Squares Positive (or negative) definiteness of a square matrix can also be characterized by some conditions about the determinants. Theorem 59 (Thm 16.1, p.382) . Let A be an m × m matrix. Then (a) A is positive definite if and only if all its m leading principal minors are positive, i.e. | A k | > , k = 1 , ··· , m . (b) A is negative definite if and only if its m leading principal minors alternate in sign as follows (- 1 ) k | A k | > , k = 1 , ··· , m . (c) If some kth order leading principal minor of A is nonzero but does not fit either of the above two sign patterns, then A is indefinite. • Theorem 16.1 is based on another method of diagonalizing quadratic forms called the technique of completing the squares ....
View Full Document

This note was uploaded on 03/26/2012 for the course ECON 205 taught by Professor Mr.lee during the Spring '11 term at Korea University.

Page1 / 4

math-2011-note12 - Definition 36(p.161 An m × m matrix A =...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online