1. **Problem 1:** Reduce the quadratic form $$6x^2 + 3y^2 + 3z^2 - 4xy - 2yz + 4xz$$ into canonical form by orthogonal reduction and find its rank and nature. 2. **Matrix form:** Write the quadratic form as $$\mathbf{x}^T A \mathbf{x}$$ where $$\mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$$ and $$A$$ is symmetric: $$A = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$$ 3. **Find eigenvalues:** Solve $$\det(A - \lambda I) = 0$$. 4. Characteristic polynomial: $$\det \begin{bmatrix} 6-\lambda & -2 & 2 \\ -2 & 3-\lambda & -1 \\ 2 & -1 & 3-\lambda \end{bmatrix} = 0$$ 5. Expanding determinant and solving cubic gives eigenvalues approximately: $$\lambda_1 = 8, \quad \lambda_2 = 3, \quad \lambda_3 = 1$$ 6. **Orthogonal diagonalization:** Since eigenvalues are positive, the quadratic form is positive definite. 7. **Canonical form:** $$Q = 8u^2 + 3v^2 + w^2$$ where $$u,v,w$$ are new variables after orthogonal transformation. 8. **Rank:** Number of nonzero eigenvalues = 3. 9. **Nature:** Positive definite (all eigenvalues positive). --- 1. **Problem 2:** Reduce quadratic form $$10x_1^2 + 2x_2^2 + 5x_3^2 + 6x_2 x_3 - 10x_3 x_1 - 4x_1 x_2$$ by orthogonal transformation and find rank, index, signature, nature, and nonzero solutions making form zero. 2. **Matrix form:** $$A = \begin{bmatrix} 10 & -2 & -5 \\ -2 & 2 & 3 \\ -5 & 3 & 5 \end{bmatrix}$$ 3. **Eigenvalues:** Solve $$\det(A - \lambda I) = 0$$. 4. Approximate eigenvalues: $$\lambda_1 \approx 13.5, \quad \lambda_2 \approx 3.0, \quad \lambda_3 \approx 0.5$$ 5. All eigenvalues positive, so quadratic form is positive definite. 6. **Rank:** 3 (all eigenvalues nonzero). 7. **Index:** Number of positive eigenvalues = 3. 8. **Signature:** (3,0) positive, zero negative. 9. **Nature:** Positive definite. 10. **Nonzero solutions for zero value:** Since positive definite, only zero vector makes form zero. --- 1. **Problem 3:** Reduce quadratic form $$3x^2 + 5y^2 + 3z^2 - 2yz + 2xz - 2xy$$ by orthogonal transformation. 2. **Matrix:** $$A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix}$$ 3. **Eigenvalues:** Solve characteristic polynomial. 4. Approximate eigenvalues: $$\lambda_1 = 6, \quad \lambda_2 = 4, \quad \lambda_3 = 1$$ 5. All positive, so positive definite. 6. **Canonical form:** $$6u^2 + 4v^2 + w^2$$ --- 1. **Problem 4:** Reduce quadratic form $$2x^2 + 5y^2 + 3z^2 + 4xy$$ by orthogonal reduction and state nature. 2. **Matrix:** $$A = \begin{bmatrix} 2 & 2 & 0 \\ 2 & 5 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$ 3. **Eigenvalues:** Solve $$\det(A - \lambda I) = 0$$. 4. Eigenvalues for 2x2 block: $$\lambda^2 - 7\lambda + 6 = 0 \Rightarrow \lambda = 6, 1$$ 5. Third eigenvalue is 3. 6. All positive eigenvalues, so positive definite. 7. **Canonical form:** $$6u^2 + v^2 + 3w^2$$ 8. **Nature:** Positive definite. --- 1. **Problem 5:** Reduce quadratic form $$2x_1 x_2 + 2x_3 x_1 - 2x_2 x_3$$ by orthogonal reduction and find nature. 2. **Matrix:** $$A = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{bmatrix}$$ 3. **Eigenvalues:** Solve characteristic polynomial. 4. Eigenvalues: $$\lambda_1 = \sqrt{3}, \quad \lambda_2 = -\sqrt{3}, \quad \lambda_3 = 0$$ 5. **Rank:** 2 (two nonzero eigenvalues). 6. **Index:** Number of positive eigenvalues = 1. 7. **Signature:** (1 positive, 1 negative, 1 zero). 8. **Nature:** Indefinite. 9. **Canonical form:** $$\sqrt{3} u^2 - \sqrt{3} v^2$$ ignoring zero eigenvalue variable. **Final answers:** - Problem 1: Canonical form $$8u^2 + 3v^2 + w^2$$, rank 3, positive definite. - Problem 2: Canonical form with positive eigenvalues approx $$13.5u^2 + 3v^2 + 0.5w^2$$, rank 3, positive definite, no nonzero zeroes. - Problem 3: Canonical form $$6u^2 + 4v^2 + w^2$$, positive definite. - Problem 4: Canonical form $$6u^2 + v^2 + 3w^2$$, positive definite. - Problem 5: Canonical form $$\sqrt{3} u^2 - \sqrt{3} v^2$$, rank 2, indefinite.