Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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9answers
2k views

Are most matrices invertible? [duplicate]

I asked myself this question, to which I think the answer is "yes". One reason would be that an invertible matrix has infinitely many options for its determinant (except $0$), whereas a non-invertible ...
17
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12answers
3k views

Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
15
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1answer
659 views

Adjoint functors as “conceptual inverses”

The Stanford Encyclopedia of Philosophy's article on category theory claims that adjoint functors can be thought of as "conceptual inverses" of each other. For example, the forgetful functor "ought ...
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4answers
927 views

Is $A + A^{-1}$ always invertible?

Let $A$ be an invertible matrix. Then is $A + A^{-1}$ invertible for any $A$? I have a hunch that it's false, but can't really find a way to prove it. If you give a counterexample, could you please ...
11
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5answers
3k views

What's the difference between arccos(x) and sec(x)

My question might sound dumb, but I don't really see why the graphics of arccos(x) and sec(x) are different, because as far as I know arccos is the inverse cosine function (cos(x)^-1) and sec equals ...
10
votes
1answer
221 views

From matrices to bipartite graphs

Assume $G(A,B)$ is a bipartite graph and assume $L(G)$ is the adjacency matrix of its line graph. define $$B=[3\text{I}+L(G)]^{-1}$$. Is it always the case that for each edge $e=(a,b)\in G$, we have: ...
9
votes
2answers
19k views

Transpose of inverse vs inverse of transpose

I can't seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix? Thanks!
9
votes
2answers
359 views

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 ...
9
votes
2answers
329 views

On the convexity of element-wise norm 1 of the inverse

Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as $$ \|A\|_1= \sum_{i,j} |A_{i,j}|. $$ Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
8
votes
6answers
2k views

If $A^2$ is invertible, then $A$ is also invertible?

True or False: If $A^2$ is invertible, then $A$ is also invertible. ($A$ is a matrix here.) The answer is true. I was trying to come up with an example that makes this false. But I couldn't. ...
8
votes
5answers
444 views

Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist? Why?

Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist, given that $\mathbf A$ is a square and positive (and possibly singular) matrix and $\epsilon$ is a small positive number? I want to use this to ...
8
votes
1answer
338 views

Simple to state yet tricky question

Define $$A=\left[\mathrm I+\sum_{k=1}^{m_1}v_k v_k^T+\sum_{k=1}^{m_2}u_k u_k^T\right]^{-1},$$ where each $u_k$ and $v_k$ is a $0$-$1$ column vector, and for each $1\leq i \leq n$, the $i$th component ...
8
votes
1answer
2k views

Inverse of a block matrix

I have a special case where $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ and: $X$ is non-singular $A$ is singular $B$ is full column rank $C$ is full row rank How do you ...
8
votes
1answer
214 views

Given the inverse of a block matrix - Complete problem

Given $X$ a block matrix $$\pmatrix{A&B}$$ where $A$ is $m \times n$ and $B$ is $m \times (nāˆ’m)$. I know a priori the value of $X \times (X^{T} \times X)^{-1}$. Substituting $X$: ...
8
votes
1answer
897 views

Functions whose derivative is the inverse of that function

Everyone knows that there are at least three functions whose derivative is the function itself, namely $e^x, \ 0$ and $-e^{x}$. ( are there more?) I was drawing some polynomials and their ...
8
votes
1answer
264 views

$\operatorname{arsinh}$ vs $\operatorname{arcsinh}$

I note that some people like to write the inverse hyperbolic functions not with the prefix "arc" (like regular inverse trigonometric functions), but rather "ar". This is because the prefix "arc" (for ...
7
votes
5answers
6k views

Prove that if $AB$ is invertible then $B$ is invertible.

I know this proof is short but a bit tricky. So I suppose that $AB$ is invertible then $(AB)^{-1}$ exists. We also know $(AB)^{-1}=B^{-1}A^{-1}$. If we let $C=(B^{-1}A^{-1}A)$ then by the invertible ...
7
votes
3answers
328 views

Is there an inverse to Stirling's approximation?

The factorial function cannot have an inverse, $0!$ and $1!$ having the same value. However, Stirling's approximation of the factorial $x! \sim x^xe^{-x}\sqrt{2\pi x}$ does not have this problem, and ...
7
votes
1answer
2k views

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 ...
7
votes
1answer
103 views

Calculating $\text{erf}^{-1}(z)$ for $z\in\mathbb{C}$

All the information I found about inverse error function $\text{erf}^{-1}(z)$ was about $z\in\mathbb{R}$. Also I found some Taylor expansions for it, but as the function is unbounded near $z=\pm1$, ...
6
votes
3answers
549 views

If $A$ and$ I+AB$ are invertible, show $I+BA$ is also invertible

Show that if $A$ and $I+AB$ are invertible, then $I+BA$ is also invertible with $$(I+BA)^{-1} = A^{-1}(I+AB)^{-1}A$$
6
votes
3answers
120 views

Multiplicative inverse of $0$

If I'm not mistaken, in a ring with identity, the additive identity cannot have a multiplicative inverse. I'm trying to prove this. Here's my attempt so far: Suppose $0\cdot a=1$ $$0\cdot a=1$$ ...
6
votes
4answers
533 views

Proof: if the graphs of $y=f(x)$ and $y=f^{-1}(x)$ intersect, they do so on the line $y=x$

This came out of a textbook problem, and as Lubin pointed out below, it's not actually true as originally stated. I'm guessing it should be restated as: If the graphs of $y=f(x)$ and $y=f^{-1}(x)$ ...
6
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2answers
496 views

Diffeomorphism from Inverse function theorem

I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
6
votes
1answer
411 views

Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...
6
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1answer
77 views

Some questions about the pseudoinverse of a matrix

For every mxn-matrix A with real entries, there exist a unique nxm-matrix B, also with real entries, such that $$ABA = A$$ $$BAB = B$$ $$AB = (AB)^T$$ $$BA = (BA)^T$$ B is called the pseudoinverse ...
6
votes
1answer
103 views

My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible

Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial. Here's my ...
6
votes
3answers
268 views

Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
5
votes
6answers
317 views

What is the inverse function of $\ x^2+x$?

I think the title says it all; I'm looking for the inverse function of $\ x^2+x$, and I have no idea how to do it. I thought maybe you could use the quadratic equation or something. I would be ...
5
votes
3answers
125 views

Can a matrix A with the property $A=A^{-1}$ only have the eigenvalues -1 and 1?

If a matrix A has the property $A=A^{-1}$, are the only possible eigenvalues 1 and -1 ? How can the matrices with integer values and the property $A=A^{-1}$ be characterized ? I found out that if ...
5
votes
2answers
714 views

Help finding inverse of $f(x)=\frac{e^x-e^{-x}}{2}$

I'm trying to find the inverse of $f(x)=\frac{e^x-e^{-x}}{2}$. My textbook says $f^{-1}(x)=\ln(x+\sqrt{x^2+1})$, but I haven't been able to get that answer. Switching $x$ and $y$, I tried solving for ...
5
votes
3answers
714 views

Inverse of symmetric matrix $M = A A^\top$

I have a matrix, generated by the product of a non-square matrix with its own transpose: $$M = A A^\top.$$ I need the inverse of $M$, assuming $\det(M) \neq 0$. Given the nature of the matrix $M$, ...
5
votes
3answers
2k views

Proving that a right (or left) inverse of a square matrix is unique using only basic matrix operations

Proving that a right (or left) inverse of a square matrix is unique using only basic matrix operations -- i.e. without any reference to higher-order matters like rank, vector spaces or whatever ( ...
5
votes
2answers
117 views

Inverse and derivative of a function [duplicate]

Find an example of an inverse function f(x) such that its derivative is the same as its inverse. I tried many different functions but non of them worked.
5
votes
3answers
394 views

Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$

I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given: $$y=W(e^{ax+b})-W(e^{cx+d})+zx$$ where $W$ is the Lambert $W$ function and ...
5
votes
1answer
119 views

Is there a name for this type of inverse?

I have a function $f : A \to B$ and an inverse $f^{-1} : B \to A$, and the only property of the inverse is that $(f \circ f^{-1} \circ f)(x) = f(x)$. In particular, it is not necessarily true that ...
5
votes
4answers
465 views

Showing a function is bijective and finding its inverse

The function f: ā„2-> ā„2 is defined by f(x,y)=(2x+3y,x+2y). Show that f is bijective and find its inverse. I've got so far: Bijective = 1-1 and onto. 1-1 if (2x1+3y1,x1+2y1)=(2x2+3y2,x2+2y2) Then ...
5
votes
2answers
427 views

Right Inverse for Surjective Function

Prove that if $f:X\to Y$ is a surjective function between sets, then there must exist a function $g:Y\rightarrow X$ such that $f\circ g=1_Y$. I know that the identity function is onto, and if $f$ ...
5
votes
2answers
144 views

How to formally show that $f(z)$ is analytic at $z=0$?

Let $z$ be a complex number. Let $$f(z)=\dfrac{1}{\frac{1}{z}+\ln(\frac{1}{z})}.$$ How to formally show that $f(z)$ is analytic at $z=0$? I know that for small $z$ we have ...
5
votes
1answer
51 views

Finding the inverse of a map from $CP^1$ to $S^2$

Given the map: $$f:CP^1 \to S^2\ ,\ f[z:w] = \left(\frac{2\mbox{Re}(w\bar{z})}{|w|^2+|z|^2},\frac{2\mbox{Im}(w\bar{z})}{|w|^2+|z|^2}, \frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$ How would I go about ...
5
votes
3answers
304 views

Find inverse for the closed-form expression of linear recurrence relation

I am trying to find an inverse of the following formula: $$ a_n=\frac{2+\sqrt{6}}{4}(1+\sqrt{6})^n+\frac{2-\sqrt{6}}{4}(1-\sqrt{6})^n $$ This formula is a closed-form expression of a linear ...
5
votes
1answer
84 views

How to find inverse of $\sin(x) + \sin(2x) = y$?

I was wondering if there were any way to solve the equation $$\sin(x) + \sin(2x) = y$$ in terms of $x$. This of course would allow us to express the inverse for this function on $-\frac{\pi}{4}$ to ...
4
votes
4answers
608 views

Is every self-inverse matrix diagonalizable?

If $A=A^{-1}$, is there always a matrix C such that $C^{-1}AC$ is a diagonal matrix (containing only -1 and 1 in the main diagonal) ? How can I check with PARI/GP, if a given matrix is ...
4
votes
7answers
142 views

How should I understand $f^{-1}(E):=\{x\in A:f(x)\in E\}$?

I understand the concept, but I still can't figure out how to read the notation: $$f^{-1}(E):=\{x\in A:f(x)\in E\}$$ I understood the concept due to the examples, not with the notation. Can someone ...
4
votes
2answers
183 views

Is $a^{-1} + b^{-1} = (a + b)^{-1}$ always true for Abelian group?

I get the equation $a^{-1} + b^{-1} = (a + b)^{-1}$ from ordinary + operation. For ordinary + operation I mean $a^{-1} = -a$. It is also true for * of rational numbers $3^{-1}*4^{-1} = \frac{1}{3} * ...
4
votes
3answers
234 views

How long does it take to consume the same amount of food

For a group of 32 students food lasts for 45 days. For how many days will the same food last for 72 students?
4
votes
3answers
387 views

Is the inverse operation on matrices distributive?

For example, is the following true: $$(A + B)^{-1} = A^{-1} + B^{-1}$$ If $\det(A) \ne 0$, $\det(B) \ne 0$, and $\det(A + B) \ne 0$.
4
votes
4answers
781 views

How to find inverse of the function $f(x)=\sin(x)\ln(x)$

My friend asked me to solve it, but I can't. If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$? I have no idea how to find the solution. I try to find ...
4
votes
2answers
365 views

What can be said about a matrix which is both symmetric and orthogonal?

I tried to find matrices A, which are both orthogonal and symmetric, this means $A=A^{-1}=A^T$. I only found very special examples like I, -I or the matrix $$\begin{pmatrix} 0 &0& -1\\ ...
4
votes
3answers
720 views

How to invert this exponential function to solve for x: $y = a \exp(bx) + c \exp(dx)$?

Cheers. So if I don't make sense, I have a value for $y$, I need to know what $x$ is. $$y = a \exp(bx) + c \exp(dx)$$ $a = 12.85$, $b = 0.001857$, $c = -54.24$, $d = -0.05316$