Problem 10
If \(A\) is positive definite, show that \(A=C^{2}\) where \(C\) is positive definite.
Problem 10
If \(U\) is a subspace of \(\mathbb{R}^{n}\), show that \(\operatorname{proj}_{U} \mathbf{x}=\mathbf{x}\) for all \(\mathbf{x}\) in \(U\)
Problem 10
A bilinear form \(\beta\) on \(\mathbb{R}^{n}\) is a function that assigns to every pair \(\mathbf{x}, \mathbf{y}\) of columns in \(\mathbb{R}^{n}\) a number \(\beta(\mathbf{x}, \mathbf{y})\) in such a way that $$ \begin{array}{l} \beta(r \mathbf{x}+s \mathbf{y}, \mathbf{z})=r \beta(\mathbf{x}, \mathbf{z})+s \beta(\mathbf{y}, \mathbf{z}) \\ \beta(\mathbf{x}, r \mathbf{y}+s \mathbf{z})=r \beta(\mathbf{x}, \mathbf{z})+s \beta(\mathbf{x}, \mathbf{z}) \end{array} $$ for all \(\mathbf{x}, \mathbf{y}, \mathbf{z}\) in \(\mathbb{R}^{n}\) and \(r, s\) in \(\mathbb{R} .\) If \(\beta(\mathbf{x}, \mathbf{y})=\beta(\mathbf{y}, \mathbf{x})\) for all \(\mathbf{x}, \mathbf{y}, \beta\) is called symmetric. a. If \(\beta\) is a bilinear form, show that an \(n \times n\) matrix \(A\) exists such that \(\beta(\mathbf{x}, \mathbf{y})=\mathbf{x}^{T} A \mathbf{y}\) for all \(\mathbf{x}, \mathbf{y}\). b. Show that \(A\) is uniquely determined by \(\beta\). c. Show that \(\beta\) is symmetric if and only if \(A=A^{T}\).
Problem 10
In each case find new variables \(y_{1}\) and \(y_{2}\) that diagonalize the quadratic form \(q\). a. \(q=x_{1}^{2}+6 x_{1} x_{2}+x_{2}^{2}\) b. \(q=x_{1}^{2}+4 x_{1} x_{2}-2 x_{2}^{2}\)
Problem 10
Find an SVD for \(A=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]\).
Problem 11
a. If a binary linear \((n, 2)\) -code corrects one error, show that \(n \geq 5\). [Hint: Hamming bound.] b. Find a (5,2) -code that corrects one error.
Problem 11
If \(A=U \Sigma V^{T}\) is an SVD for \(A,\) find an SVD for \(A^{T}\).
Problem 11
a. Show that \(A\) is hermitian if and only if \(\bar{A}=A^{T}\). b. Show that the diagonal entries of any hermitian matrix are real.
Problem 11
Let \(A\) be a positive definite matrix. If \(a\) is a real number, show that \(a A\) is positive definite if and only if \(a>0\).
Problem 11
If \(U\) is a subspace of \(\mathbb{R}^{n},\) show that \(\mathbf{x}=\operatorname{proj}_{U} \mathbf{x}+\operatorname{proj}_{U^{\perp}} \mathbf{x}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{n}\)
