1.II .7C. 7 \mathrm{C} \quad

Vectors and Matrices | Part IA, 2008

Prove that any nn orthonormal vectors in Rn\mathbb{R}^{n} form a basis for Rn\mathbb{R}^{n}.

Let AA be a real symmetric n×nn \times n matrix with nn orthonormal eigenvectors ei\mathbf{e}_{i} and corresponding eigenvalues λi\lambda_{i}. Obtain coefficients aia_{i} such that

x=iaiei\mathbf{x}=\sum_{i} a_{i} \mathbf{e}_{i}

is a solution to the equation

Axμx=f,A \mathbf{x}-\mu \mathbf{x}=\mathbf{f},

where f\mathbf{f} is a given vector and μ\mu is a given scalar that is not an eigenvalue of AA.

How would your answer differ if μ=λ1\mu=\lambda_{1} ?

Find aia_{i} and hence x\mathbf{x} when

A=(210120003) and f=(123)A=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right) \quad \text { and } \quad \mathbf{f}=\left(\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right)

in the cases (i) μ=2\mu=2 and (ii) μ=1\mu=1.

Typos? Please submit corrections to this page on GitHub.