Intereting Posts

Evaluating Stratonovich integral from definition
Why the $O(t^2)$ part in $L(t) = L + t(\csc \alpha_i – \cot \alpha_i +\csc \alpha_{i+1} – \cot \alpha_{i+1}) + O(t^2)$?
Sum of nil right ideals
The resemblance between Mordell's theorem and Dirichlet's unit theorem
expansion of $\text{ cosh}(z+1/z)$
Find a formula for a sequence $\{\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{3\sqrt{3\sqrt{3}}},…\}$
formal proof from calulus
Prove $\lim_{n\to\infty}x_n=2$ Given $\lim_{n \to \infty} x_n^{x_n} = 4$
Examples of problems that are easier in the infinite case than in the finite case.
If $\lambda_n \sim \mu_n$, is it true that $\sum \exp(-\lambda_n x) \sim \sum \exp(-\mu_n x)$ as $x \to 0$?
Square free finite abelian group is cyclic
If you roll a fair six sided die twice, what's the probability that you get the same number both times?
Cheap proof that the Sorgenfrey line is normal?
Understanding the process of finding eigenvectors
outcome from fair and non-fair dice, dice are chosen from other dice part2

The inverse of a non-singular lower triangular matrix is lower triangular.

Construct a proof of this fact as follows. Suppose that $L$ is a non-singular lower triangular matrix. If $b \in \mathbb{R^n}$ is such that $b_i = 0$ for $i = 1, . . . , k \leq n$, and $y$ solves $Ly = b$, then $y_i = 0$ for

$i = 1, . . . , k \leq n$. (Hint: partition $L$ by the first $k$ rows and columns.)

Can someone tell me what exactly we are showing here and why it will prove that the inverse of **any** non-singular lower triangular matrix is lower triangular?

- Linearly independent functionals
- What does “isomorphic” mean in linear algebra?
- Questins on Formulae for Eigenvalues & Eigenvectors of any 2 by 2 Matrix
- Is it true that $ U \oplus W_1 = U \oplus W_2 \implies W_1 = W_2 $?
- Signed angle between 2 vectors?
- Linear algebra - Dimension theorem.

- Why is $\det(A - \lambda I)$ zero?
- Finding a coefficient of a unknown to have a unique solution in a system
- Similarity of matrices over $\Bbb{Z}$
- Interchanging Rows Of Matrix Changes Sign Of Determinants!
- Given a matrix, is there always another matrix which commutes with it?
- What does it mean to represent a number in term of a $2\times2$ matrix?
- How does linear algebra help with computer science
- Formula for cylinder
- What is the relation between analytical Fourier transform and DFT?
- How to solve following logarithmic equation: $n(n-1)3^{n} = 91854$

Let’s write $$L^{-1}=[y_1\:\cdots\:y_n],$$ where each $y_k$ is an $n\times 1$ matrix.

Now, by definition, $$LL^{-1}=I=[e_1\:\cdots\:e_n],$$ where $e_k$ is the $n\times 1$ matrix with a $1$ in the $k$th row and $0$s everywhere else. Observe, though, that $$LL^{-1}=L[y_1\:\cdots\:y_n]=[Ly_1\:\cdots\: Ly_n],$$ so $$Ly_k=e_k\qquad(1\leq k\leq n)$$

By the proposition, since $e_k$ has only $0$s above the $k$th row and $L$ is lower triangular and $Ly_k=e_k$, then $y_k$ has only $0$s above the $k$th row. This is true for all $1\leq k\leq n$, so since $$L^{-1}=[y_1\:\cdots\:y_n],$$ then $L^{-1}$ is lower triangular, too.

$$********$$

Here’s an alternative (but related) approach.

Observe that a lower triangular matrix is nonsingular if and only if it has all nonzero entries on the diagonal. Let’s proceed by induction on $n$. The base case ($n=1$) is simple, as all scalars are trivially “lower triangular”. Now, let’s suppose that all nonsingular $n\times n$ lower triangular matrices have lower triangular inverses, and let $A$ be any nonsingular $(n+1)\times(n+1)$ lower triangular matrix. In block form, then, we have $$A=\left[\begin{array}{c|c}L & 0_n\\\hline x^T & \alpha\end{array}\right],$$ where $L$ is a nonsingular $n\times n$ lower triangular matrix, $0_n$ is the $n\times 1$ matrix of $0$s, $x$ is some $n\times 1$ matrix, and $\alpha$ is some nonzero scalar. (Can you see why this is true?) Now, in compatible block form, we have $$A^{-1}=\left[\begin{array}{c|c}M & b\\\hline y^T & \beta\end{array}\right],$$ where $M$ is an $n\times n$ matrix, $b,y$ are $n\times 1$ matrices, and $\beta$ some scalar. Letting $I_n$ and $I_{n+1}$ denote the $n\times n$ and $(n+1)\times(n+1)$ identity matrices, respectively, we have $$I_{n+1}=\left[\begin{array}{c|c}I_n & 0_n\\\hline 0_n^T & 1\end{array}\right].$$ Hence, $$\left[\begin{array}{c|c}I_n & 0_n\\\hline 0_n^T & 1\end{array}\right]=I_{n+1}=A^{-1}A=\left[\begin{array}{c|c}ML+by^T & M0_n+b\alpha\\\hline x^TM+\alpha y^T & y^T0_n+\beta\alpha\end{array}\right]=\left[\begin{array}{c|c}ML+by^T & \alpha b\\\hline x^TM+\alpha y^T & \beta\alpha\end{array}\right].$$ Since $\alpha$ is a nonzero scalar and $\alpha b=0_n$, then we must have $b=0_n$. Thus, $$A^{-1}=\left[\begin{array}{c|c}M & 0_n\\\hline y^T & \beta\end{array}\right],$$ and $$\left[\begin{array}{c|c}I_n & 0_n\\\hline 0_n^T & 1\end{array}\right]=\left[\begin{array}{c|c}ML & 0_n\\\hline x^TM+\alpha y^T & \beta\alpha\end{array}\right].$$ Since $ML=I_n$, then $M=L^{-1}$, and by inductive hypothesis, we have that $M$ is then lower triangular. Therefore, $$A^{-1}=\left[\begin{array}{c|c}M & 0_n\\\hline y^T & \beta\end{array}\right]$$ is lower triangular, too, as desired.

Suppose you have an invertible lower-triangular matrix $L$. To find its inverse, you must solve the matrix equation $LX = I$, where $I$ denotes the $n$-by-$n$ identity matrix.

Based on how matrix multiplication works, the $i^{\text{th}}$ column of $LX$ is equal to $L$ times the $i^{\text{th}}$ column of $X$. In order for $LX = I$, it must be that the first $i-1$ entries in the $i^{\text{th}}$ column of $LX$ are all zero. The hint is that you can prove that this implies that the first $i-1$ entries in the $i^{\text{th}}$ column of $X$ must all be zero. To do this, you can explicitly write out your calculation, using your assumption that $L$ is lower-triangular. You’ll get a fairly easy system of linear equations to analyze.

In simple form, we can write A = D*(I+L); where A is lower triangular matrix, D is diagonal matrix, I is identity matrix and L is lower triangular with all zeros in diagonal.

Since $A^{-1} = (I+L)^{-1}*D^{-1}$ and inverse of D is simply inverse of diagonal element. And for very large n $L^{-n} = 0$ since it is having only lower triangular elements.

And we can write $(I+L)^{-1} = I – L + L^2 – L^3 + …. (-1)^n*L^n$ which itself is lower triangular matrix.

- Mathematical equivalent of Feynman's Lectures on Physics?
- An elegant way to solve $\frac {\sqrt3 – 1}{\sin x} + \frac {\sqrt3 + 1}{\cos x} = 4\sqrt2 $
- If $q^k n^2$ is an odd perfect number with Euler prime $q$, which of the following relationships between $q^2$ and $n$ hold?
- Prove conjecture $a_{n+1}>a_{n}$ if $a_{n+1}=a+\frac{n}{a_{n}}$
- Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$
- Factorials, Simplify the addition and multiplication of factorials.
- Find the values of $p$ such that $\left( \frac{7}{p} \right )= 1$ (Legendre Symbol)
- Why is the decimal representation of $\frac17$ “cyclical”?
- Finding moment generating functions for a dice roll
- Evaluate $\int_{0}^{\infty} \frac{{(1+x)}^{-n}}{\log^2 x+\pi^2} \ dx, \space n\ge1$
- Is the set of closed points of a $k$-scheme of finite type dense?
- Why does drawing $\square$ mean the end of a proof?
- Is the condition “sample paths are continuous” an appropriate part of the “characterization” of the Wiener process?
- I'm trying to find the longest consecutive set of composite numbers
- Kelly criterion with more than two outcomes