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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Functions that are their own inversion.

What are the functions that are their own inverse? (thus functions where $ f(f(x)) = x $ for a large domain) I always thought there were only 4: $f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ ...
25
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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 ...
22
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5answers
44k views

How to find the inverse modulo m?

For example: $$7x \equiv 1 \pmod{31} $$ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do? Update If I have ...
19
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11answers
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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 ...
19
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8answers
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Why do negative exponents work the way they do? [closed]

Why is a value with a negative exponent equal to the multiplicative inverse but with a positive exponent? $$a^{-b} = \frac{1}{a^b}$$
17
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6answers
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What's the inverse operation of exponents?

You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the ...
16
votes
2answers
38k 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!
15
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1answer
739 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 ...
14
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4answers
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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
7k 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 ...
11
votes
3answers
164 views

Existence of continuous angle function $\theta:S^1\to\mathbb{R}$

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all ...
10
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6answers
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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. ...
10
votes
5answers
593 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 ...
10
votes
1answer
3k 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 ...
10
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1answer
242 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
7answers
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Why is the inverse of a sum of matrices not the sum of their inverses?

Suppose $A + B$ is invertible, then is it true that $(A + B)^{-1} = A^{-1} + B^{-1}$? I know the answer is no, but don't get why.
9
votes
2answers
478 views

Why aren't integration and differentiation inverses of each other?

Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral: $$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int ...
9
votes
3answers
809 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
9
votes
2answers
974 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
416 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
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5answers
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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 ...
8
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4answers
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Matrix inverses - Why are they derived the way they are?

Note that this is not a question of how, but why. I know the mechanics of it, but this is the first thing i've come across that truly seems like magic, rather than a rigorous mathematical process. ...
8
votes
4answers
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Calculating the Modular Multiplicative Inverse without all those strange looking symbols

I am sure all those symbols are really easy for you guys to understand, but I would appreciate it if someone could bring it down to earth for me. How could I do this on a basic calculator? or with a ...
8
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4answers
133 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
8
votes
1answer
398 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
229 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 $m \times (n−m)$. I know a priori the value of $X \times (X^{T} \times X)^{-1}$. Substituting $X$: ...
8
votes
1answer
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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
275 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 ...
8
votes
1answer
95 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
7
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3answers
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Why are nonsquare matrices not invertible?

I have a theoretical question. Why are non-square matrices not invertible? I am running into a lot of doubts like this in my introductory study of linear algebra.
7
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3answers
426 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
3answers
739 views

How to derive compositions of trigonometric and inverse trigonometric functions?

To prove: $$\begin{align} \sin({\arccos{x}})&=\sqrt{1-x^2}\\ \cos{\arcsin{x}}&=\sqrt{1-x^2}\\ \sin{\arctan{x}}&=\frac{x}{\sqrt{1+x^2}}\\ \cos{\arctan{x}}&=\frac{1}{\sqrt{1+x^2}}\\ ...
7
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1answer
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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 ...
7
votes
1answer
327 views

Can A be singular? [duplicate]

Let $A\in \mathbb{C}^{n\times n}$ satisfy $$A^{2}+A+I=0 $$ Can A be singular? So I have: $$ (A-I)(A^{2}+A+I)=0\\ A^{3} = I \\ (\det A^{3}) = \det(I) \\ (\det A)^{3} = 1\\ \det A\neq 0 $$ So $A$ is ...
7
votes
2answers
699 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 ...
7
votes
2answers
940 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 $f$ ...
7
votes
1answer
127 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
143 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 ...
6
votes
3answers
629 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
167 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
2answers
633 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\\ ...
6
votes
3answers
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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 $M$, ...
6
votes
4answers
675 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
votes
1answer
96 views

If $f(AB) =f(A)f(B)$, then $A$ is inversible iff $f(A)\neq 0$

Let $f:\mathscr M_n(\mathbb K) \to \mathbb K$ be a non constant function such as $f(AB) = f(A)f(B)$ for all $A,B$ in $\mathscr M_n(\mathbb K)$. The question is to show that $M\in GL_n(\mathbb K)$ iff ...
6
votes
2answers
159 views

Efficient way to compute $(A+D)^{-1}$ when $A^{-1}$ is known

I need to compute the inverse of a matrix sum $A+D$, where the inverse of $A\in\mathbb{R}^{n\times n}$ is known. The matrix $D\in\mathbb{R}^{n\times n}$ is a diagonal matrix which can be thought of as ...
6
votes
1answer
54 views

Show that $Y$ is invertible

Let X be a $40\times40$ matrix such that $X^3 = 2I$. I want to show that $Y= X^2 -2X + 2I$ is invertible as well. I tried working with the equations to see if I can get Y as a product of matrices ...
6
votes
1answer
618 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
votes
1answer
61 views

Tan inverse summation

$$S=\sum\limits_{i=1}^{4}\tan^{-1} x_i$$ How to simplify this ? I think I will have to use this : but it looks too long a method . Is there a method or symmetrical way which yields ...
6
votes
1answer
120 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
143 views

Moore-Penrose Pseudo-inverse of a matrix on adding 1 new row/column

Given that I know the pseudo-inverse of a matrix(not necessarily a square matrix), how to calculate the pseudo-inverse of the matrix I get by adding a single row/column to the original matrix? i.e, ...