Quadratic forms arise often in geometry, physics and engineering and it is desirable to reduce these to canonical (standard) forms. Reduce the quadratic form:Q = 3x12 + 5 x22 + 3x32 - 2x2 x3 + 2x1 x3 - 2 x1 x2 to a standard form by making an approximate change in variables X = MY , where M is an orthogonal matrix. i) Write Q as X T AX . ii) Find the Eigenvalues of A. Is the quadratic form...
Reduce the following quadratic forms to canonical form by means of a modal matrix; are they positive definite? (a) F(x1,2) 4x -2 (b) F(x,2, 3) x-812 + 4x13 (c) F(, 2, 3) = 2r13223
XTAX=1. determine their canon- 1. Write the following quadratic forms as V(x) ical forms, find the modal matrices (i.e. the matrices of unit eigenvectors) of the corresponding transformations and write down explicite expressions for canonical cOordinates (y1, 2, y3) in terms of the original coordinates (x1, X2, X3). State what surfaces these quadratic forms correspond to = > (a) x + 4x1r2 + 4a13-8a2x3 = 1; (b) a3a3a^ + 4xj2 +4x131223 1; (c) 4a7 2a2 2axjx2 2x13+ 6x23 = 1....
5. [2 marks] Write down the relation matrix of the abelian group Now reduce this matrix using elementary integer row and column operations, and hence write G as a direct sum of cyclic groups. 5. [2 marks] Write down the relation matrix of the abelian group Now reduce this matrix using elementary integer row and column operations, and hence write G as a direct sum of cyclic groups.
Consider the quadratic form Q(1, 2, r)2r2r34rs. Write Q(, 2, 3) in the fornm Q(1, 2, z3)xAx for some matrix A to be found, where x-2 T3 Classify Q(x1, r2, r3) as positive definite, negative definite, positive semidefinite, negative semidefinite, or indefinite
Prove that type 1 elementary matrix is a product of type 2 and 3 elementary matrices
1. [8 marks] Write equation 7x{ +6x1x2 + 7x3 = 1 as a matrix-vector quadratic form, convert it to a canonical form and determine the type of a curve to which it corresponds. 2. [16 marks] Find the spectral matrix and the corresponding modal matrix for -5 0 157 B = 3 4 -9. Write down the formula that needs to be used to diago- -5 0 15 nalise matrix B, but do not perform matrix multiplications.
4) a) For the system of equations given, partially row reduce the coefficient matrix in the following careful way: X1 + 2yı - 2 = 5 4x+9y, - 32 = 8 (5x + 12yı - 324 = 1 Stage 1: just reduce the matrix first to an upper triangular form U and leave pivot entries as they are (don't multiple to change them to 1's). Reduce from left to right through the columns and from the pivot entry down within...
4) a) For the system of equations given, partially row reduce the coefficient matrix in the following careful way: *1 + 2y, - 2 = 5 4x1 +9y1 - 32 = 8 (5x + 12y - 321 = 1 Stage 1: just reduce the matrix first to an upper triangular form U and leave pivot entries as they are (don't multiple to change them to 1's). Reduce from left to right through the columns and from the pivot entry down...
need help a) For the system of equations given, partially row reduce the coefficient matrix in the following careful way: *1 + 2yı - 24 = 5 4x1 +9yı - 321 = 8 (5x, +12yı - 324 = 1 Stage 1: just reduce the matrix first to an upper triangular form U and leave pivot entries as they are (don't multiple to change them to l's). Reduce from left to right through the columns and from the pivot entry down...