Homework 9: Linear Algebra

Problem 1
 

Find an example of a square matrix \( A \) with real entries such that \( A^2=0 \) but \( AA^T\neq 0 \).

Problem 2
 

Assume that \( A \) is an \( n\times n \) matrix with real entries \( a_{ij} \) for \( i,j\in\{1,2,\dots n\} \) such that \( a_{ij}=0 \) if \( i\geq j \). Prove that \( A^n=0 \).

Problem 3
 

Assume that \( A \) and \( B \) are two quadratic matrices such that \( AB=BA \). Prove that if \( x \) is an eigenvector for \( A \), then \( Bx \) is an eigenvector for \( A \) as well.

Problem 4
 

Solve the system of recursive equations: \begin{eqnarray*} a_{n+1}&=& 2a_n+2b_n\\ b_{n+1}&=& 3a_n + c_n\\ c_{n+1}&=& -3a_n+c_n, \end{eqnarray*} with the initial conditions \( a_0=2 \), \( b_0=-1 \), \( c_0=1 \).

Problem 5
 

Let \( A \) and \( B \) be linear transformations on a finite dimensional vector space \( V \). Prove that \( \dim \ker (AB)\leq \dim\ker (A)+\dim\ker (B) \).

\( \ker (A) \) denotes the kernel of \( A \), i.e. \( \ker (A)=\{v: A(v)=0\} \), and \( \dim V \) denotes the dimension of the vector space \( V \).

Problem 6
 

Given an \( n\times n \) matrix \( M \) with complex entries, let us denote by \( M^* \) its conjugate transpose defined as \( M^*=\overline{M}^T \). The matrix \( M \) is called normal if \( AA^*=A^*A \). A matrix \( U \) is called unitary if \( U^*=U^{-1} \).

  • (a) Prove that if \( M \) is any complex matrix, then there exists a unitary matrix \( U \) and an upper-triangular matrix \( D \) such that \( M=UDU^* \).

  • (b) Prove that if \( M \) is a normal matrix, then \( D \) from part (a) is diagonal.

Problem 7
 

Does there exist an \( n\times n \) real matrix \( A \) such that \( \mbox{tr }(A)=0 \) and \( A^2+A^T=I \)? (\( \mbox{tr }(A) \) is the trace of the matrix \( A \), and \( A^T \) is its transpose).

Problem 8
 

Let \( \delta> 0 \) be given. Assume that \( (A_n)_{n=1}^{\infty} \) is a sequence of subsets of a unit cube such that the volume of each \( A_n \) is greater than or equal to \( \delta \). Prove that for each \( \varepsilon> 0 \) there are integers \( i \) and \( j \) such that the volume of \( A_i\cap A_j \) is at least \( \delta^2-\varepsilon \).


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