Lecture Notes

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====== Sample Midterm 2 Problems ======

==== Computational ====

1. Consider the following Hermitian 2x2 matrices $Q$, find an invertible matrix $A$ and diagonal matrices $D$, such that $Q = A^* D A$.
  * $ Q = \begin{bmatrix} 1 & 2 \cr 2 & 2  \end{bmatrix} $
  * $ Q = \begin{bmatrix} 0 & i \cr -i & 0  \end{bmatrix} $
  * $ Q = \begin{bmatrix} 0 & i \cr -i & 2  \end{bmatrix} $

2. Find the eigenvalues of the following matrix, and for each eigenvalue find an eigenvector. 
$$ T =  \begin{bmatrix} 1 & 2 & 5 \cr 0 & 2 & 3 \cr 0 & 0 & -1  \end{bmatrix} $$

3. Let $A$ be the following $3 \times 3$ matrix, Use Gauss Elimination to find $\det A$ and $A^{-1}$
$$ A =  \begin{bmatrix} 0 & 2 & -1 \cr 1 & 2 & 1 \cr 0 & 3 & -1  \end{bmatrix} $$

4. Let $v_1 = (0, 2, 1)$ , $v_2 = (1, 2, 3)$, and $v_3 = (1,1,1)$. Let $V_*$ denote the complete flag associated to $v_i$, namely $V_1 = span(v_1), V_2 = span(v_1, v_2), V_3 = span(v_1,v_2,v_3)$. Find a linear transformation $A$ on $\R^3$, that take the flag $V_*$ to the standard flag (i.e the flag associated to the standard basis $e_1,e_2,e_3$). 

Conceptual

5. True or False
  * Every quadratic form on $\R^n$, under a change of coordinate, can be written as $$ X_1^2+\cdots + X_r^2 - X_{r+1}^2 - \cdots - X_{r+s}^2$$ for some $r \geq 0, s \geq 0$ with $r+s \leq n$. 
  * Every Hermitian form on $\C^n$, under a change of coordinate, can be written as $$ X_1^2+\cdots + X_r^2 - X_{r+1}^2 - \cdots - X_{r+s}^2$$ for some $r \geq 0, s \geq 0$ with $r+s \leq n$. 
  * For every quadratic form $Q$ on $\R^n$, there exists an orthonormal basis $\{e_i\}$ (with respect to the standard inner product on $\R^n$), such that $Q(e_i, e_j)=0$ for $i\neq j$. 
  * For every Hermitian form $H$ on $\C^n$, there exists an orthonormal basis $\{e_i\}$ (with respect to the standard Hermitian inner product on $\C^n$), such that $H(e_i, e_j)=0$ for $i\neq j$. 
  * For any linear transformation $T: \R^n \to \R^n$, we can find $n$ eigenvalues (possibly with repetition) $\lambda_1, \cdots, \lambda_n$, and corresponding eigenvectors $v_1, \cdots, v_n \in \R^n$, such that $v_i$ forms a basis, and $T (v_i) = \lambda_i v_i$. 
  * Let $Q_1, Q_2$ be two quadratic form on $\R^n$, is $Q_1 + Q_2$ also a quadratic form? 
  * Let $Q_1$ be the standard quadratic form on $\R^n$, $Q_2$ be any quadratic form on $\R^n$. Can one find an basis $e_1, \cdots, e_n$ that is orthogonal with respect to both $Q_1, Q_2$? 

==== Others =====
For the application of the Sylvester rule, one can refer to the homework question.
Lecture Notes

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