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Questions tagged [inverse]

Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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Invert a large sized matrix

I have a big matrix of size 200000 x 200000. I need to compute its inverse. But it gives out of memory error. Is there any algorithm to approximate and compute the inverse of a large sized matrix.
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How to invert this type of infinite series?

If have a function $f$ given by a series $$ f(z) = \sum_{n,m = 0}^\infty u_{n,m} z^{n + m t} $$ for some $t\in\mathbb{R}^+$. Is there an straightforward way (something similar to the inversion ...
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1answer
27 views

Finding the derivative of inverse of the product of matrices

I need to calculate the following derivative of the product of several matrices (one of which is the inverse of a product of matrices) with respect to one of the matrices in question: $$\frac{\delta(\...
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3answers
30 views

Give a 2*2 block matrix $M = \begin{bmatrix}A&B\\0&C \end{bmatrix}$ and find a formula for $M^{-1}$ in terms of $A$, $B$, and $C$

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section2.5 Problem 29, I was puzzled at solving it. Here is the problem description: Give a 2*2 ...
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1answer
55 views

How is the inverse of $y=4x^3 - 3x^4$ found?

I would like to calculate the inverse of $y = 4x^3 - 3x^4$ on the domain $x = [0,1]$. What would be the best way to tackle this? I'd preferably a general method, suitable for tackling other ...
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2answers
39 views

Fourier transform of exponential function

It is a function $A(f,t)= e^{j 2 \pi (t/a-af)}$ I would like to take the inverse Fourier transform. So: $B(\tau,t)=\int A(f,t) e^{j2\pi f\tau}df=\delta(\tau-t_0)...???$ How to solve this integral? ...
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1answer
20 views

Invert a function containing integration term

Consider this simple equation $$ \tau(t)=\int\frac{dt}{a(t)},$$ where $\tau$ and $a$ are functions of $t$. Now, from this equation, how can I calculate $a(\tau)$ ?
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1answer
34 views

Finding a difficult inverse Laplacetransform

I'm trying to solve the following problem: $$\mathcal{L}_\text{s}^{-1}\left[\frac{\text{F}(\text{s})+\text{G}(\text{s})}{1-\exp\left(-\frac{\text{s}}{4x}\right)}\right]_{\left(t\right)}\tag1$$ Where:...
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There exists a continuous inverse of $(\text{id}-A)$ in the set $(\text{id}-A)(X)$.

Exercise : Let $X$ be a Banach space and $A \in \mathcal{L}_c(X)$ (means that $A$ is a compact operator). Suppose that $(\text{id}-A)$ is $"1-1"$. Show that the operator $(\text{id}-A)$ has a ...
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Using Gauss-Jordan to invert an upper triangular matrix

Let $$U = \begin{bmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{bmatrix}$$ where $a, b, c \in \mathbb R$. Use Gauss-Jordan elimination ...
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1answer
23 views

Finding the left and right inverses of block matrices

Let $\bf{A_{11}}$ $\in$ $\mathbb{F}^{pxp}$, $\bf{A_{12}}$ $\in$ $\mathbb{F}^{pxq}$, and $\bf{A_{21}}$ $\in$ $\mathbb{F}^{qxp}$. Show that if $\bf{A_{11}}$ is invertible, then a) The block matrix [$\...
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1answer
34 views

Sum of two positive denfinite matrices invertible — where is my mistake?

I wrote the following statement: $A(t) = \sum_{i \le t} r(i) r(i)^\top + \alpha I_N$ where $r(i) \in \mathbb{R}^N $. As a sum of positive definite matrices $r(i)r(i)^\top$ and a ...
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24 views

Is the inverse of the Jacobian equivalent to the Jacobian of the inverse?

$ \widetilde \rho = \left [ \begin{matrix} \rho & \theta & \phi \\ \end{matrix} \right ]^\top \; $ and $ \widetilde x = \left [ \begin{matrix} x & y & z \\ \end{...
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1answer
83 views

How can I show that this matrix has no inverse?

Let $A = [a_{ij}]$ be an $n \times n$ matrix with entries in $\mathbb{R}$. Suppose there exists an $m$ with $a_{ij} = 0$ for $i \ge m$ and $j \le m$, and $a_{i,i} \ne 0$ for $1 \le i \lt m$. Show that ...
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Factorization of Square-integrable random-variables and Generalized Inverses

Suppose that $X,Y,Z \in L^2(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^d)$, are $d$-dimensional random-vectors and there exists functions $f,g\in L^2(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),Law(X);\...
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1answer
39 views

Prove that $A^{-1} + B^{-1}$ nonsingular by showing that $(A^{-1} + B^{-1} )^{-1} = A( A + B )^{-1} B$

Let $A$, $B$, and $A + B$ be nonsingular matrices. Prove that $A^{-1} + B^{-1}$ is nonsingular by showing that $( A^{-1} + B^{-1} )^{-1} = A( A + B )^{-1} B$ I have done progress to only knowing ...
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Let $G,G'$ be two digraphs, show that $\phi^{-1}$ is an isomorphism

Problem So let $\phi : G \rightarrow G'$ be an isomorphism between two directed graphs. Prove that $\phi^{-1}$ is an isomorphism. Also prove if $H \leq Aut(G')$ then $\phi^{-1}H\phi\leq Aut(G)$. My ...
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3answers
65 views

What does the notation $A^{-2}$ mean if $A$ is a matrix?

$A^2$ means to multiple the matrix by itself, and $A^{-1}$ refers to the matrix's inverse. Would $A^{-2}$ be the square of the inverse or the inverse of the square?
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2answers
32 views

Does polynomial keep inverse?

Let $A=(a_{i,j})_{n\times n}$ be an invertible matrix with the positive rational entry. Let $p(x)$ be a rational polynomial. Consider the following matrix \begin{align*} B=\left(p(a_{i,j})\right)_{n\...
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1answer
20 views

Inverse Laplace with a irreducible quadratic in the denominator and a 1 in the numerator

Please help me find the Inverse Laplace transform of: $$F(s) = \frac{1}{s(s^2+8s+4)}$$ After completing the square, I obtained $$F(s) = \frac{1}{s((s+4)^2-12)}$$ Thank you.
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1answer
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Finding the inverse of this isomorphic function.

I need to find the the inverse of the isomorphic function $f:\Bbb R^3 \rightarrow \Bbb R^3$ given by $\begin{pmatrix}a\\b\\c\end{pmatrix} \rightarrow \begin{pmatrix}3b-a\\3a+c\\3b-c\end{pmatrix}$ ...
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1answer
136 views

Is there such a thing in math the inverse of a sequence?

Such as can I construct a sequence by reversing the order of the approximating sequence of $\frac{1}{3}$? So such inverse would look like $\left\{….,0.333333333,0.3333333,0.333333,...0.3\right\}$. I ...
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Inverse of differential operator and boundary conditions

I want to clear a point that "Why boundary conditions are important in taking inverse of any differential operator (lets say Laplace operator )?". What i understood is that any transformation is ...
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1answer
56 views

Solution to a modified linear system using two methods

I am trying to obtain the solution to a modified linear system. I am comparing two methods to solve this modified linear system, and I'm noticing some issues with one of the methods. A linear system ...
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2answers
38 views

Show that $2$ is invertible in a ring with odd cardinality. [closed]

Can somebody explain me how to prove that $2$ is invertible in a commutative ring with an odd number of elements, please? I've found that $2 \ne 0$ from Lagrange.
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0answers
21 views

Inverse of a matrix which is difference of a singular matrix with a small diagonal matrix?

If $A$ is a real symmetric singular matrix (similar to a Laplacian matrix, which comes from M'GM, where M is incidence matrix and G is a diagonal matrix) with large values and $B$ is a diagonal matrix ...
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2answers
22 views

Finding set of solutions to $AX=0$ for triangular matrix $A$.

Let $\ A = (a_{ij})_{i,j=1}^n$ be a triangular n × n-matrix, that is, $\ a_{i,j}=0$ if $1 ≤ j < i ≤ n$. Assume none of diagonal elements is equal to zero. Find the set of all solutions $X\in\Bbb{R}^...
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37 views

Movement of $1/z$ in complex plane

How is moving $\frac{1}{z}$ in complex plane if $z$ is described by a circle which has radius $r$ and center $a+b*i$ I've just started complex algebra and still having some trouble imagining it. How ...
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2answers
39 views

Invertibility of a block matrix

I need to prove that the following matrix is invertible $$\left( {\begin{array}{*{20}{c}} {{B_{n \times n}}}&{{I_{n \times m}}}\\ {{I_{m \times n}}}&{{0_{m \times m}}} \end{array}} \right),$$ ...
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1answer
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Solve for the inverse of $\mathbf I - \tan(\frac{\phi}{2}) \mathbf {\hat \omega}$

Original problem comes from some notes on rotations (at the last page), which was devoted to deriving Rodrigues' rotation formula. The complete problem is to show why $$(\mathbf I - \tan(\frac{\phi}{2}...
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2answers
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Inverse of $v \cdot v^\top$

Let's say I have a vector $v$. Now I want to calculate $(v\cdot v^\top)^{-1}$. Is there a known formula to solve this more directly than simply calculating it directly? Maybe something similar to the ...
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Existence of striclty monotone transformation

Assume we have a function class $F$ containing bivariate functions $f(x,y)\; (f: \mathcal{X} \times \mathcal{Y} \to \mathbb{R})\ $ that are continuously differentiable with respect to each argument. ...
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1answer
34 views

What is a big condition number for a matrix?

The condition number of a matrix is a measure of how close a matrix is to being singular. But, what is considered a big condition number?
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59 views

Finding $A^{-3}$ using Cayley Hamilton Theorem

If $$A = \begin{bmatrix} 2 & 4 \\ 1 & 1 \\ \end{bmatrix}, $$ then use the Cayley-Hamilton Theorem to find $A^{-3}$. This is how far I have gotten: \begin{align} p(\lambda) &= \lambda^2 -...
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proving a diagonally dominant matrix is invertable using banach lemma?

Here how they solve the problem, however there are some problem understanding their conclusion. I know by banach lemma if we prove that norm $B$ is less than $1$ then $B+I$ matrix is invertible. My ...
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1answer
54 views

First-year matrix problem: How do you show that a sum of an identity matrix and another matrix is equal to the sum's inverse?

The following is the problem at hand: $A^4 = 2A^2.$ Prove that $(I-A^2) = (I-A^2)^{-1}$ My attempts at a solution: $I = A^{-1} * A,$ therefore we can start with $(A^{-1}A - A^2) = (A^{-1}A - (1/2)A^...
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How to find the inverse or a tight bound on a series

If $ f(x)=1-\frac{4}{\pi}\sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1}e^{-\frac{\pi^2 (2k+1)^2}{8 x^2}}$, find $\min\{x:f(x)\geq 1-y \}$. The function $f(x)$ is increasing and its output falls in $[0,1]$...
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18 views

Inverse of $(rP+\bar{r}\bar{P})$ where $P=P^*$ and $r=\exp(i\theta)$

I have a positive definite Hermitian matrix $P=P^*>0$ where $P^*$ is the conjugate transpose of $P$ and $r=\exp(i\theta)$. So, how can I prove that $$ (rP+\bar{r}\bar{P})^{-1}=rY+\bar{r}\bar{Y}$$ ...
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1answer
43 views

Why doesn't $AA^-A=A \Rightarrow A^-AA^-=A^-$?

The condition for being a generalized inverse matrix is $AA^-A=A$. There's another condition $A^-AA^-=A^-$, and when this also holds, $A^-$ is called a reflexive inverse. But when does it happen ...
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1answer
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Inverse for a sum of two matrices

What is the inverse of the 3x3 matrix $(\vec{a} \vec{a}+cI)^{-1}$ Where $\vec{a} \vec{a} $ is a dyad and $I$ is the identity matrix, $c$ is constant, $\vec{a}=(a_1,a_2,a_3)$ is a 3x1 vector $\vec{a}...
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1answer
54 views

Is infinity the reciprocal of zero/is zero the reciprocal of infinity?

Is infinity the reciprocal of zero? Is zero the reciprocal of infinity? It would make sense that they would be--they behave in a similar way (anything multiplied by zero or infinity results in zero or ...
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2answers
52 views

Does 3x4 matrix have an Inverse? Why? [duplicate]

I saw this question somewhere and made me think do 3x4 matrices have an inverse, as I previously that that only square matrices have an inverse. If non-square matrices have an inverse, especially if ...
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4answers
66 views

Can you please solve $7^{-1} \mod 480$ using extended Euclidean Algorithm?. Kindly show the steps till end

I am solving RSA algorithm wherein I have to find d by finding $7$ inverse modulo $480$. Please help in solving till end using extended euclidean algorithm Using extended Euclidean Algorithm for ...
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12 views

Conditions for inverse to be continuous in parameters

Let $f(x,\theta,\omega):X \ \times \ \Theta \ \times \ \Omega \to \mathbb{R}$, where $X\subset\mathbb{R}, \ \Theta\subset\mathbb{R}, \text{and} \ \Omega\subset\mathbb{R}_{++}$ are intervals, be ...
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53 views

Inverse of $2 \times 2$ block matrix

I find this simple equation, curious about how the last two equalities derived? Can anybody share some insights about this?
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1answer
18 views

vector transpose multiplied by matrix then multiplied again by the same vector, find vector

Suppose w is a 1-d vector, X is a d by d positive semi-definite symmetric matrix, and a is a constant. Is there a way to express w in terms of X, a? $$ w^T X w = a^2 $$ I was thinking something like $...
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1answer
38 views

Solutions to $1/K!+1/L!+1/M!=1/N!$

Is there more than one solution to $\frac{1}{K!}+\frac{1}{L!}+\frac{1}{M!}=\frac{1}{N!}$ where $K, L, M, N$ are all natural numbers? The one solution i came up with was to assume that $K=L=M$, ...
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If we have set of observations (having multiple factors) , how can we find mathematical relationship between them

If we have set of observations (having multiple factors x,y,z) , What is the best way to find mathematical relationship between them. Secondly, how can the mathematical relationship /equation found ...
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1answer
18 views

If $f:X\to Y$ is bijective, then $(f^{-1})^{(-1)}(A)=f(A)$

Prove that if $f:X\to Y$ is bijective, then $(f^{-1})^{(-1)}(A)=f(A)$. Here, $^{-1}$ stands for inverse while $^{(-1)}$ stands for preimage. My trial Since $f:X\to Y$ is bijective, $f^{-1}:Y\to X$ ...
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0answers
41 views

Inverse Fourier transform of complex fourier transform

How can i find the inverse fourier transform when the fourier transform is given in complex form the module $$|x(\omega)|=|\omega|$$ and the $$\phi(\omega)=-3\omega$$ should i find the inverse ...