# 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.

3,029 questions
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### How does this matrix math with complex numbers work/where is the mistake.

I am a beginner to these types of math, so you will have to forgive me if this question has an obvious answer. I just learned how to get the inverse of a matrix, so I thought it would be interesting ...
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### Inverse function to the exponential integral

$Ei(-ln(f(x)) = c + ln(ln(x))$ where $Ei(x)=$ - $\int_{-x}^{\infty} \frac{e^{-t}}{t} dt$, where $x$ be a non zero real (called exponential integral ). Is there a way to express $f(x)$? If yes, ...
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### Finding the inverse of function for $x>0$

I have the following function $$f(x) = \frac{1}{2x}\left(\sigma(x) - \frac{1}{2}\right)$$ where $\sigma(x)$ is the sigmoid function $$\sigma(x) = \frac{e^x}{1 + e^x} = \frac{1}{1+e^{-x}}$$ I ...
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### Given to find $\arccos\left(\cos(\frac{14 \pi}{3})\right)$

Okay so $\arccos\left(\cos\left(\dfrac{14 \pi}{3}\right)\right)$ can be written as $\arccos\left(\cos\left(4 \pi+\dfrac{2 \pi}{3}\right)\right)$ yielding answer as $\frac{2\pi}{3}$ But why can't we ...
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### How to prove that in an abelian group $-(-a) = a$?

I have to prove that in an abelian group $-(-a) = a$, and the only hint given is that the inverses are unique. My attempt is as follows: $-(-a) = a$ is equivalent to $-(-a) - a = 0$, but I don't ...
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### Let $A$ be an $n*n$ matrix such that $A^3=A^2+A-I$. If $A$ Is diagonalizable Show that $A=A^{-1}$

Let $A$ be an $n*n$ matrix such that $A^3=A^2+A-I$. Show that $A$ is invertible Suppose in $A$ is diagonalizable. Show that $A=A^{-1}$ For the first part I managed to do it by a ...
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### Inverse of an Elementary Matrix

Assume we have a 3x3 matrix like: A = 9 8 7 6 5 4 3 2 1 We are applying an Elementary matrix E to A: ...
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### Calculating a matrix-vector product efficiently without inverse operation

Given a matrix $A=I+K^{-1}BB^TK^{-1}$ where K is a discrete Laplacian, and B is a sparse uniformly distributed random matrix, how can I calculate matrix-vector products $Av$ efficiently without any ...
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### 2D convolution with Gaussian using Fourier transform

I was solving 2D diffusion equation with initial condition \chi(x,0)=1 in the circle centered at origin with radius r. Equation I want to solve To solve this equation efficiently, I need to use ...
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### Compute inverse of ill-conditioned matrix

I need to compute inverse of a matrix that is highly ill conditioned and nearly singular. I tried using Jacobi preconditioning, a method to add a scalar value to the diagonal entries of the original ...
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### Finding inverse operator (Green function)

I'm working on some quantum field theory and have to operate on a field with the following operator: $$(x^\mu \partial_\mu + 1)^{-1}$$ I've been trying to find an explicit form of this operator, ...
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### Guaranteed invertible matrix

Let it be two $m \times n$ matrices: $A$ and $B$, where $m,n \geq2$. Rows of these matrices are linearly independent. So, which matrix is guaranteed invertible: $AA^T$, $B^TB$, $AB^T$, $A^TB$? I ...
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### Pseudo inverse of a singular matrix without any linearly independent rows or columns

It is given here that "when A has linearly independent columns (and thus matrix $A^{*}A$ is invertible), $A^{+}$ can be computed as: $A^{+}=(A^{*}A)^{-1}A^{*}$ " and there is a similar expression for ...
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### Linear Algebra Matrix Skew Symmetric [closed]

Recall that an nxn matrix is called skew-symmetric if A^T=-A. a)Prove that for all x that are in R^n we have x^TAx=0 (note: x^T Ax is a scalar for any nxn matrix A) b) Prove that I+A is invertible My ...
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### Linear Algebra Inverse [duplicate]

Let $𝐴$ be an $𝑛×𝑛$ matrix and $𝑂$ be the zero $𝑛×𝑛$ matrix. a) Suppose that $𝐴^2=𝑂$. Prove that $𝐼+𝐴$ is invertible. b) Suppose that $𝐴^𝑘=𝑂$ for some $𝑘$. Prove that $𝐼+𝐴$ is ...
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### Given that $A+B$ is invertible, prove or disprove $A(A+B)B = B(A+B)A$ as well as show that $A(A + B)^{-1}B = B(A + B)^{-1}A$

I have tried coming up with a counterexample for the first one, but it has worked each time so my intuition is that the first statement is true. I've tried using the fact that $\det(A+B)$ does not ...
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### Need help understanding an equation: composition, addition and inverse

I have found an interesting paper on a digital image registration algorithm. There are many equations in the paper that I only understand partially, but there is a particular one I would like to ...
1answer
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### Inverse function of $ax + bx^3$

I am trying to find the inverse of the function $y = f(x) = ax + bx^3$, i.e. $x = f^{-1}(y)$. (The equation arises in the modeling of a certain type of transmission used in robots) Looking at the ...
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### Elements reduction in a general case

If $G$ is a group, then each element has an inverse and $\forall x, y, z \in G, xy = xz \Rightarrow x^{-1} \cdot xy = x^{-1} \cdot xz \Rightarrow 1 \cdot y = 1 \cdot z \Rightarrow y = z$ However, we ...
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### Invertibility of the Schur complement when $D=0$

Suppose we have a partitioned matrix $$X = \begin{bmatrix} A & B \\ C & O\end{bmatrix}$$ where $O$ is a zero matrix of proper dimensions and where $B$ and $C$ are nonsquare matrices. Also ...
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### What's inverse function of f(x,y) = (2x+3y,3x+2y)

The question is in the title - assuming that f: Q X Q -> Q X Q, what's inverse function of f(x,y) = (2x+3y,3x+2y)? For the life of me, I just can't figure it out.
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### Proving a matrix identity

Let $M \in \mathbb{R}^{n\times n}$ with $\|M\| < 1$. Show $$(I - M)^{-1} = I + M(I - M)^{-1}.$$ How can I do this? I tried starting with the equality $$(I - M)(I - M)^{-1} = I,$$ Then I ...
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### How to show that the following 2 matrices are conjugate?

How to show that the following 2 matrices are conjugate? \begin{bmatrix} z & 0 \\ 0 & z^{-1} \end{bmatrix} And \begin{bmatrix} z^{-1}& 0 \\ 0 & z \end{bmatrix} I know the ...
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### diagonal matrix and invertible matrix proof [closed]

I am given the following proof question: Let $A \in {\mathbb R}^{n\times n}$.` Show that there exist invertible matrices $B$, $C$ such that $A=B+C$. I believe it has something to do with ...