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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Reverse of matrix multiplication when matrix is non-invertible

In factor analysis, a random variable can be expressed as: $$ \mu(y) = \beta^TX+\epsilon \\ Var(y) = \beta\Sigma\beta^T+Cov(\epsilon) $$ where $\beta \in R^{N \times M}$ is factor loadings and $\Sigma ...
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Unique left inverse of linear transformation must also be a right inverse

A linear transformation $T$ has at most one left inverse. If $L$ is a left inverse, must it also be a right inverse? I have proved that, when the transformation $T: V \rightarrow W$ has a finite ...
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Proving a matrix in invertible from the following question

The question says: If matrix $A$ is invertible and $A + B = AB$, prove that matrix $B$ is invertible and $A^{-1} + B^{-1} = I$ Firstly, I was thinking that we can only prove that matrix $B$ is a ...
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Combining two linear matrix equations with non-square linear operators?

Assume I have two equations: $$\mathbf{x} = \mathbf{A}\mathbf{y}$$ where $\mathbf{x} \in \mathbb{R}^{m}$, $\mathbf{y} \in \mathbb{R}^{n}$, and $\mathbf{A}$ is a matrix of size $m$-by-$n$, as well as $$...
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Inverse of $\psi(t,s)=(e^t,se^{2t})$

I am trying to solve a problem involving finding the inverse of this map $$\psi:\mathbb{R}^2\to \mathbb{R}^2$$ $$\psi(t,s)=(e^t,se^{2t})$$ Here is what I am thinking Let $x=e^t$ and $y=se^{2t}$. ...
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How can I find a matrix $A∈\mathbb{R}^{2\times2} $ for which ${A}^{3} =\begin{bmatrix} 64 & 64 \\ -9 & 16\end{bmatrix}^{-1}$? [closed]

My question is to find a matrix $A∈\mathbb{R}^{2\times2}$ for which $${A}^{3} =\begin{bmatrix} 64 & 64 \\ -9 & 16\end{bmatrix}^{-1}.$$ Thanks.
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1answer
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How can I find matrix x which has cos(x) value [closed]

I want to find matrix x which has cos(x) value. I tried to solve with this equation ax+bI=cos(x). with cos^-1(A)=X but doesnt work. Find a matrix X ∈ ℝ2 x 2 for which cos(x)= $$ \begin{matrix} 1 &...
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Chain rule question for derivative of inverse

I feel like I'm missing something obvious here. I understand the graphical proof of $f^{-1'}[a] = {1\over f{'}[f^{-1}(a)]}$, but I'm stuck on one aspect of the simple non-graphical proof. Start with $...
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Prove it or give a counterexample to the statement [closed]

Prove it or give a counterexample to the statement If $A$ is an invertible $n\times n$ matrix, then the number of solutions of $Ax = b$ depends on the vector $b \in\mathbb R^n$.
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Invertibility of Mixing Matrix $M$ in $A=CMR$

I'm interested in solving the following problem from Strang's Linear Algebra and Learning from Data: If $C$ and $R$ contain bases for the column space and row space of $A$, why does $A=CMR$ for ...
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Inverting a sum of Kronecker's delta

I am trying to evaluate the following expression: \begin{equation} \begin{split} A=\sum_{\ell_1\ell_2\ell_3}\sum_{\ell_1'\ell_2'\ell_3'}C_{\ell_1\ell_2\ell_3}\Big[&\delta_{\ell_1\ell_1'}\delta_{\...
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Inverse of the sum of matrices

I have two square matrices: $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-...
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Show that $(I-\mathbf x\mathbf y^T)^{-1} = I-\frac{1}{\mathbf x^T\mathbf y- 1}\mathbf x\mathbf y^T$

Let $x,y\in\mathbb R^n$ and suppose that $x^Ty \neq 1$. Show that $(I-\mathbf x\mathbf y^T)^{-1} = I-\frac{1}{\mathbf x^T\mathbf y- 1}\mathbf x\mathbf y^T$. Note, I need to compute this directly not ...
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Inverse of a Toeplitz matrix with FFT-based methods

I have a covariance matrix $Q$ and need to find $Q^{-1}$. Here, $Q$ is a Toeplitz matrix. I want to calculate the inverse of the matrix with FFT-based methods rather than the conventional ones like ...
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1answer
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A question about Inverse Matrix

I have a question about Inverse Matrix. I would appreciate if anyone could provide some help. Question: Suppose that a matrix A satisfies the equation $A^2-4A+3I=0$. Find an expression for $A^-1$. ...
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If $A^2+A=0$, is A invertible? [closed]

As the title states, if $A^2+A=0$, is it possible to determine whether A is invertible? This's how I'm thinking. Factoring we get: $A(A+I)=0$ Hence, either $A=0$ or $A=-I$ We know that $A = 0$ ...
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How Can I solve for a vector by multiplying the matrix inverse for non invertible matrix?

I have the followingenter image description here equation I need to find {N} what I did is : multiply both sides by [X1 X2 ....]' then I multiply both side by the inverse of the multiplication inv([X1 ...
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1answer
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What to do if the Extended Euclidean Algorithm terminates in one step?

I am trying to solve the linear congruence $14x \equiv 1 \pmod{113}$. So I first find $\gcd(14, 113) = 1$. However this means that: $113 = 14(8) + 1$ There is only one step needed. If I don't have ...
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3answers
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More than one modular multiplicative inverse possible?

I am redoing exams as a preparation and I found this weird particular exercise to me. "Does $32$ have a multiplicative inverse in modulo $77$? If yes, calculate the inverse." Since the $\gcd(77,32)$...
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4answers
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Derivative of inverse function $\sin^{-1}(x)^2$

So $y=\sin^{-1}(x)^2$ I am asked to find $\frac{dy}{dx}$ Using the chain rule I find $\frac{dy}{dx}$= $2\sin^{-1}(x) * \frac{d}{dx}(\sin^{-1}(x))$ I let $z = \sin^{-1}(x)$ Multiplying both ...
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Pseudo Inverse of Jacobian Matrix

I have a script written by someone else that outputs end-effector velocities. I need to transform these end-effector velocities to joint velocities. This requires the pseudo-inverse of the Jacobian ...
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1answer
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How can you simplify/verify this solution for $\int\limits_0^{.25991…} Q^{-1}(x,x,x)dx?$

As I do not know the complex behavior of this function, it would be even harder to integrate past the real domain. The upper bound for the domain is a constant I will denote β. $${{Q_2}=\int_0^βQ^{-1}(...
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Approximate the inversion of a matrix based on the norm relationship?

Assume: $$ \left\| \mathrm{A} \right\| _2\gg \left\| \mathrm{B} \right\| _2 $$ where $A$ and $B$ are square and non-singular. Can we approximate the inverse of $(A+B)$ by inverse of $A$? In other ...
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Calculating the trace of the product of two matrices

I have to calculated $\mbox{trace}(A^{-1}B)$ where $A$ is a symmetric positive definite matrix and $B$ is a symmetric matrix, very sparse with only two elements non zero. I want to find a way that I ...
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How do you proceed when constructing a homeomorphism? [closed]

I have maybe quite stupid and quite methodological question: how do you proceed when constructing homeomorphism? I am asking, because I came across many exercises, proofs etc., where some steps would ...
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2answers
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Is the trace of inverse matrix convex?

Hi I would like to know whether the trace of the inverse of a symmetric positive definite matrix $\mathrm{trace}(S^{-1})$ is convex. Actually I know that the trace of a symmetric positive definite ...
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1answer
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Inverse function $f(x) = 3^x$

$f(x) = 3^x$ find $x $ when $f^{-1}(x) = 4$ $y = 3^x$ $x=log_{3}(y)$ $\therefore f^{-1}(x) = log_{3}(x)$ $4= log_{3}(x)$ $x = 3^4$ $x = 81$ simple solution but I was helping my little brother with his ...
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On completing the solution for $\int_0^1 Q^{-1}(x,x) dx$ and other constants.

$\Large{\text{Introduction:}}$ Here is a link to the Inverse Regularized Incomplete Gamma function used in this problem. For simplicity, let the unit interval be expressed as $I=[0,1]$: $$\mathfrak{Q}=...
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Inverse of $A-I$

I am trying to understand one detail of one solution of below question Let $A$ be n-by-n matrix ($n \geq 2$ ), and $$ A= \begin{pmatrix} 0 & 1 & 1 & \cdots & 1 \\ 1 & 0 & 1 &...
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Multiplicative Inverse of a Power Series

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = -\frac{1}{...
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Complexity of solving equation (D+L)X=U where D, L and U are diagonal, lower triangular and upper triangular matrices respectively

I am trying to find the complexity of solving equation (D+L)X=U where D, L and U are diagonal, lower triangular and upper triangular matrices, respectively. I have used forward substitution method, ...
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Can the inverse of this matrix equation be expanded?

In my current project, I have the following expression: $$\left(A^{-1}(A^{-1})^\intercal+A^{-1}BC^{-1}(C^{-1})^\intercal B^\intercal(A^{-1})^\intercal\right)^{-1}$$ where all matrices $A$, $B$, and $C$...
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Is it possible to compute the inverse of a matrix using infinite series of matrix powers?

I am trying to see if there is a way to compute $A^{-1}$ for $A \in \mathbb{R}^{n\times n}$. I used $B=I-A$ to compute $(I-B)^{-1}$ but now I'm not sure about which conditions must be satisfied so ...
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Inverse of cosh(x)

My goal is to find the inverse of $y=\cosh(x)$ Therefore: $$x=\cosh(y)=\frac{e^y+e^{-y}}{2}=\frac{e^{2y}+1}{2e^y}$$ If we define $k=e^y$ then: $$k^2-2xk+1=0$$ $$k=e^{y}=x\pm\sqrt{x^2-1}$$ $$y=\ln(x\...
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1answer
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How to invert these matrix via Gauss method?

Im trying to find the inverse matrix based on $\left[\begin{array}{ccc} 2 & 2 & -1\\ 0 & 4 & -1\\ -1 & -2 & 1 \end{array}\right]$ $R_{1}=$ First row $R_{2}=$ Second row $R_{3}=$...
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1answer
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If $f(a)$ is invertible under a ring homomorphism $f$, is $a$ invertible too? [duplicate]

Suppose $f\colon R\to S$ is a ring homomorphism (and not rng homomorphism, wherein $f(1) = 1$ is not generally true). I can prove that if $a$ is invertible$^1$, then $f(a)$ also is, with its inverse ...
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If $(f\circ g)(x)=x$ does $(g\circ f)(x)=x$?

Given $$(f◦g)(x)=x$$ (from R to R for any x in R) does it mean that also $$(g◦f)(x)=x$$ I feel like its not true but I can't find counter example :( I tried numerous ways for several hours but I ...
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Prove explicitly that $(\alpha I +B)^{-1}x = x/\alpha$ if $Bx=0$

Assume that a matrix $B$ (real or complex, it's not important) has a null eigenvector x, namely $B x = 0$. Therefore, $B^{-1}$ does not exist (the kernel of $B$ has dimension greater or equal to 1). ...
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What is the inverse of $f(x)=\frac{\sqrt{x}}{x-1}$?

It is bijective so it should have one. I was solving some of my homework when I came upon this. The answer key had a question mark for the solution. I looked it up on Wolfram Alpha and there is a ...
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1answer
44 views

Inverse of two matrix sums

Trying to learn a basic implementation from the following post: Inverse of the sum of matrices and also from https://www.jstor.org/stable/2690437?seq=1#page_scan_tab_contents The post indicates that $$...
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trace of inverse operator

Let $H$ be an RPK-Hilbert space and $K:X\times X\rightarrow \mathbb{R} $ be the reproducing Kernel s.t. $K$ is bounded by $1$. For some Probability Space $(X, \nu)$ It is assumed that all $f \in H$ ...
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1answer
38 views

Why do we need to ensure that equation $Ax=b$ has at most one solution for each value of $b$ for the matrix $A$ to have an inverse?

For every point to have a solution in $Ax=b$ the matrix must contain at least one set of $m$ linearly independent columns. But I wonder why for the matrix to have an inverse, we additionally need to ...
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1answer
50 views

Inverse of a sum of special matrices

Does anyone know of any tricks to evaluate the following inverse of matrix sums? $$ \left( \sum_{i=1}^m D_i A D_i \right)^{-1} $$ where the $D_i$ are non-singular diagonal matrices, and $A$ is a ...
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27 views

Inverse of piecewise bijective function

Given $$f(x) = -2 x ^ 3 + 3 x ^ 2, 0 <= x <= 1$$ where I'd like to find $f^{-1}(x)$ defined on the same domain. I've tried using Mathematica where I get an answer, but it's not defined exactly ...
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28 views

Invert of a summation matrix

I am solving a FEM problem involving contacts. This problem leads to a matrix problem that I believe is commonly encountered. Assume a matrix $A$ as follows: $$A = B + C$$ where $B$ is an invertible $...
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38 views

Schur complement like operation on a singular matrix

For the classical definition of matrix inversion by Schur complement, given by: \begin{aligned} M^{-1}=\left[\begin{array}{ll} A & B \\ C & D \end{array}\right]^{-1} &=\left(\left[\begin{...
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1answer
37 views

Inverse of a Fractional Exponent?

Hello Mathematics Stack Exchange, I'm currently a Grade $11$ Math Student and to train for this year's exam I am going through a worked video of the previous years' exams to get a better understanding ...
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37 views

Gradient of the product of an inverse matrix and a vector

I would like to expand the gradient of the following product of the matrix $M$ and the vector $v$ using the product rule $$\nabla (M^{-1} \cdot v) = M^{-1} \cdot \nabla v + \nabla (M^{-1}) \cdot v$$ ...
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1answer
38 views

Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ .... Is invertibility assumed?

Given a function f: A→B where X, Y $\subset$ B, prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ I was able to go ahead and prove this statement by applying the definition of the pre-...
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1answer
103 views

Finding $\mathrm{\int_{\Bbb R} a^{cosh(x)} dx}$

After the success and great answer by @metamorphy of my $$\mathrm{\int_{-\pi}^0 a^{csc(x)}dx}$$ question, I experimented a bit more and found this nice graph. This looks almost like a gamma function ...

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