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math-2011-note12

# math-2011-note12 - Denition 36(p.161 An m m matrix A =(ai j...

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Definition 36 (p.161) . An m × m matrix A = ( a ij ) is called an upper-triangular matrix if a ij = 0 for i > j. A is called a lower-triangular matrix if a ij = 0 for i < j. A is called a diagonal matrix if a ij = 0 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 | > 0 , 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 | > 0 , 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 . 49

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Consider m = 2. F = a 11 x 2 1 + 2 a 12 x 1 x 2 + a 22 x 2 2 = a 11 ( x 1 + a 12 a 11 x 2 ) 2 + bracketleftBigg a 22 - ( a 12 ) 2 a 11 bracketrightBigg x 2 2 . Define y 1 = x 1 + a 12 a 11 x 2 , y 2 = x 2 . Then F = a 11 y 2 1 + bracketleftBigg a 22 - ( a 12 ) 2 a 11 bracketrightBigg y 2 2 .
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