Question

Let $L in mathbb{C}^{m imes m}$ be a unit lower-triangular matrix (i.e., with diagonal entries equal to 1 ). For convenience, write $L$ in the form and define $M=L^{-1}$ (a) Derive a formula for $m_{i j}$ (which may involve other entries of $M$ ). Which entries of $L$ does $m_{i j}$ depend on? (b) Suppose the subdiagonal entries of $L$ are independent random numbers ±1 with equal probability. Fix $k$ and define $mu_{1}=m_{k k}, mu_{2}=m_{k+1, k}, mu_{3}=$ $m_{k+2, k}, ldots .$ Write down a system of recurrence relations with random coefficients for the numbers $mu_{j}$ (c) Experiments show that random triangular matrices with entries ±1 are exponentially ill-conditioned in the sense that if $kappa_{m}$ denotes the 2 -norm condition number of a matrix of this kind of dimension $m,$ then $lim _{m ightarrow infty}left(kappa_{m} ight)^{1 / m}=C$ for some constant $1<C<1.5 .$ (The limit process can be made precise in various ways, but we shall not go into the technicalities; think of it as holding "with probability $1 .^{prime prime}$ ) Perform numerical experiments involving random matrices of various dimensions to estimate $C$ to $10 %$ accuracy or better. (d) Larger scale experiments become feasible if the random matrices of (c) are replaced by the random sequences $mu_{1}, mu_{2}, mu_{3}, ldots$ of (b). Explain (without proof $)$ why the constant $C$ can also be obtained by considering these sequences, and carry out numerical experiments to estimate $C$ to $1 %$ accuracy or better.

          Let $L in mathbb{C}^{m 	imes m}$ be a unit lower-triangular matrix (i.e., with diagonal entries equal to 1 ). For convenience, write $L$ in the form 
and define $M=L^{-1}$
(a) Derive a formula for $m_{i j}$ (which may involve other entries of $M$ ). Which entries of $L$ does $m_{i j}$ depend on?
(b) Suppose the subdiagonal entries of $L$ are independent random numbers
±1 with equal probability. Fix $k$ and define $mu_{1}=m_{k k}, mu_{2}=m_{k+1, k}, mu_{3}=$ $m_{k+2, k}, ldots .$ Write down a system of recurrence relations with random coefficients for the numbers $mu_{j}$
(c) Experiments show that random triangular matrices with entries ±1 are exponentially ill-conditioned in the sense that if $kappa_{m}$ denotes the 2 -norm condition number of a matrix of this kind of dimension $m,$ then $lim _{m 
ightarrow infty}left(kappa_{m}
ight)^{1 / m}=C$ for some constant $1<C<1.5 .$ (The limit process can be made precise in various ways, but we shall not go into the technicalities; think of it as holding "with probability $1 .^{prime prime}$ ) Perform numerical experiments involving random matrices of various dimensions to estimate $C$ to $10 %$ accuracy or better.
(d) Larger scale experiments become feasible if the random matrices of (c) are replaced by the random sequences $mu_{1}, mu_{2}, mu_{3}, ldots$ of (b). Explain (without proof $)$ why the constant $C$ can also be obtained by considering these sequences, and carry out numerical experiments to estimate $C$ to $1 %$ accuracy or better.

Added by Monica O.

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