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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204 views

relation between size of matrix and condition number

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
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1answer
38 views

Square root entries of matrices

How would you simplify something like this? $((\xi'\omega \xi)^{-1})^{0.5}$ where $\xi$ is a $k \times 1$ matrix, $\omega$ is a $k\times k$ square matrix. Thank you very much! Edit: Yes, though ...
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82 views

How to find the inverse of a function $f:\mathbb{Z}_{30}\to\mathbb{Z}_{30}$ defined by $f([a])=[7a]$

If $f:\mathbb{Z}_{30}\to\mathbb{Z}_{30}$ is a function defined by $f([a])=[7a]$, show that $f$ is one-to-one and onto, and find $f^{-1}$. I've got proof that the function is well defined, one to one, ...
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1answer
51 views

About non-negative matrix

If $W$ is diagonal matrix with each entry $W_{i,i}>0$, $K$ is a symmetric and positive semi-definite matrix and $K_{i,j}>0$ (actually $K$ is a kernel matrix and calculated from a RBF kernel ...
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2answers
12k 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!
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2answers
489 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 ...
6
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4answers
425 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 $y=f^{-1}(x)$ ...
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2answers
144 views

Invertible functions in $ R^m$

The definition of an invertible function that my book (Apostol's Mathematical Analysis) gives is: A function $f:S \to\mathbf{R}^n$, where $S$ is open in $\mathbf{R}^n$, has a unique inverse if $f$ is ...
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0answers
153 views

Optimization problem about large matrices

I'd like to solve the following optimization problem: Find non-negative scalar $a$, $b$, $c$ to minimize $\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$ where ...
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146 views

Low-rank approximation to the Graph Laplacian matrix of a regular grid.

As mentioned in the title, does anybody know any methods of efficient low-rank approximation $LL^T$ to the Graph Laplacian matrix $A$ corresponding to a square lattice? (except PCA)
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1k views

Determine if the matrix is idempotent?

I am dealing with an example to show that the matrix($M = I − X(X'X)^{−1}X'$) is idempotent. X is a matrix with T rows and k columns and I the unit matrix of dimension T. And then to determine the ...
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3answers
5k views

How to find the inverse of this exponential function?

Here is the function... It is $y=e^{x-3}+5$, I have no clue how to find the inverse of it. I graphed the function but now it says find the inverse and graph it. I do not know how to graph it.
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1answer
70 views

Why $ g(p) = 0.5 p^{-0.2} + 0.5 p^{-0.5} $ has a well-defined inverse that is continuous and strictly decreasing.

A book that I am reading claims the following about the function $ g(p) = 0.5 p^{-0.2} + 0.5 p^{-0.5} $ (which is a demand function): Formal arguments based on the Intermediate Value Theorem and ...
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2answers
252 views

Matrix Inverse Question

Let $C$ be an invertible 2x2 matrix such that: $$C^{-1} \cdot \begin{bmatrix}1 \\ 2\end{bmatrix} = \begin{bmatrix}3 \\ 4\end{bmatrix}$$ $$C^{-2} \cdot \begin{bmatrix}9 \\ 5\end{bmatrix} = ...
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2answers
74 views

Application of Matrix Diagonalization

I'm reading a book about inverse analysis and trying to figure out how the authors do the inversion. Assume that matrix $C$ is $$ C ~=~ \begin{bmatrix} 88.53 & -33.60 & -5.33 \\ ...
3
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1answer
70 views

How to compute $\text{trace}((A+D)^{-1}A)$

Give a diagonal perturbation matrix $D$ (which is not an identity matrix), is there a simple way to compute $$\text{trace}((A+D)^{-1}A)$$ Or is there a good approximation?
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70 views

Given the product of a unitary matrix and an orthogonal matrix, can it be easily inverted _without_ knowing these factors?

Given the product $M$ of a unitary matrix $U$ (i.e. $U^\dagger U=1$) and an orthogonal matrix $O$ (i.e. $O^TO=1$), can it be easily inverted without knowing $U$ and $O$? Sure enough, if $M=UO$, then ...
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2answers
40 views

Inverting all values in matrix

Lets say I have a matrix: $$\left[\begin{array}{cc} 2 & 4 \\ 3 & 7 \\ \end{array}\right] $$ And my maximum range value is $10$, how would I go about creating another matrix that ...
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2answers
209 views

Reverse rows in a matrix

To rotate a matrix 180 degrees around the center point, what I am planning to do is first transverse the matrix, then reverse the rows and then do it again to produce the final result. This works and ...
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1answer
43 views

When is (a restriction of) the map $y = f(x) = x + \frac{n}{x}$ bijective, if $x, y \in \mathbb{Q}$ and $n \in {\mathbb{Z}}^{+} \cup \{0\}$?

A good day to everyone. If $0 < x \in \mathbb{Q}$, $0 < y \in \mathbb{Q}$, $n \in {\mathbb{Z}}^{+} \cup \{0\}$, and $n$ is squarefree, then the function $$y = f(x) = x + \frac{n}{x}$$ is not ...
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1answer
61 views

How to invert finding marginals of a rotationally symmetric distribution?

Given $$f(x) = \int_{-\infty}^{+\infty} g(x^2+y^2) \,\mathrm{d}y,$$ is it possible to reconstruct $g(u)$ analytically? I tried differentiating $f(x)$ but only found $$f'(x) = \int_{x^2}^{+\infty} ...
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1answer
45 views

Is aI + bA invertible if rank(A) = n-1?

I am not able to prove this for sure by myself... To be more precise, $A$ is a $n \times n$ matrix of rank $n-1$ such that all diagonal elements of A are positive, off-diagonal elements can be ...
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434 views

Inverse Jacobian matrix of spherical coordinates

I found inverse transformation from spherical coordinates to cartesian coordinates (on $x>0$, $y>0$ and $z>0$). I have $$ r = w_1(x,y,z) = \sqrt{x^2+y^2+z^2} $$ $$ \theta = w_2(x,y,z) = ...
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1answer
392 views

computing the inverse of a special sparse matrix

Given a high-dimensional symmetric postive-definite matrix with only the main diagonal and several other diagonal (say, 1st, 5th and 100th) above and below the main diagonal to be non-zero and all ...
2
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2answers
146 views

How to invert sum of matrices?

Given are two matrices: $\bf A, \bf B$ We know that matrices $\bf A \neq \bf B$ are invertable, symmetric, positive-definite and of full rank. Is it possible to give the formula for following sum ...
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275 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 $\mathbb{R}^{n \times m}$. ...
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1answer
49 views

If $A$ is invertible, so is $A^*A$

Let $A \in L(H)$, for a Hilbert space $H$. If $A$ is invertible, why is $A^*A$ invertible, too?
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1answer
702 views

Inverse of orthogonal projection

I have an $N \times N$ orthogonal projection matrix $P = A^H(AA^H)^{-1}A$ that I'm trying to find the inverse for. I'm using matlab, however, I keep getting the warning "the matrix is close to ...
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51 views

Transpose of 2 matrices together

So if I have an $m\times n$ matrix $A$ and I represent that matrix as $\displaystyle A = QR$, how do I write $A^{T}$ (transpose) in terms of the original $\displaystyle QR$? Does it become ...
2
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1answer
339 views

Inversion formula for the Abel transform

I need an inversion formula for the Abel transform $$ F(y) = 2\int_y^\infty\frac{f(r)r\,dr}{\sqrt{r^2-y^2}}. $$ Hint: The inversion formula found on Wikipedia appears to be incorrect. The ...
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3answers
436 views

Inverse of sum of two functions

Assuming two functions are invertible, is it true that the inverse of the sum of the two functions is the sum of the inverses (assuming all functions are well behaved)?
2
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2answers
103 views

calculating an inverse of a split function

I am having difficulty taking the inverse of the following function: $$ f(x)=\begin{cases} \frac{1}{4 \sqrt{ |1-x|}} & \text{if} \ x\in [0,2] \\ 0 & \text{otherwise}\end{cases}$$ Could ...
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2answers
2k views

RYB and RGB color space conversion

I am working on a project where I need to convert colors defined in RGB (Red, Green, Blue) color space to RYB (Red Yellow Blue). I managed to solve converting a color from RYB to RGB space based on ...
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1answer
106 views

The inverse of the matrix $\{1/(i+j-1)\}$

Let $n$ be a positive integer. Show that the matrix $$\begin{pmatrix} 1 & 1/2 & 1/3 & \cdots & 1/n \\ 1/2 & 1/3 & 1/4 & \cdots & 1/(n+1) \\ \vdots & \vdots & ...
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1answer
431 views

Derivative of matrix inverse

I am trying to find the derivative of a matrix with respect to the inverse of the same matrix. The matrix in question is a non singular symmetric matrix. Any thoughts?
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1answer
98 views

inverse of function

Thanks for the help! Here is the solution.. i had a problem: $$f(x)=\frac{(\sqrt x+8)}{(5-\sqrt x)}$$ i had to find the inverse, so lets begin... 1) first i write in terms of $y$ ...
3
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1answer
415 views

Matrix Pseudo-Inverse using LU Decomposition?

What is the step by step numerical approach to calculate the pseudo-inverse of a matrix with M rows and N columns, using LU decomposition? So far, I have found this, but it uses singular value ...
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3answers
1k views

Evaluate the derivative of an inverse function by using a table of values?

Given the function and derivative values in the table below, evaluate $\frac{d}{dx}f^{-1}(3)$ ...
5
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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 ...
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1answer
868 views

Prove that if A is an invertible matrix, then A*A is Hermitian and positive definite.

If I'm not mistaken, if a matrix M has its conjugate M*=M, then M is Hermitian. In this case then, am I asked to show that (A*A)*=A*A ? What does it have to do with A being invertible though? And ...
3
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0answers
82 views

Inverse of a sub-matrix

I have a multivariate Gaussian distribution with known $\mu$ and $\Sigma$. I want to evaluate it given a vector $x$. However, some of the attributes of this vector may be unknown, in which case I want ...
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1answer
44 views

Proving the area function has an inverse

I am able to differentiate A at x using the FTC, but then I was wondering how one could show that A was one to one and prove that it has an inverse. If anybody could please help.
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1answer
233 views

Identify the equation of the normal line?

Identify the equation of the normal line to the curve $y=g(p)=2.5+3.5(4^p)$ where it crosses the $y$-axis. So I am guessing the normal line would be the inverse of the derivative function, since it ...
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3answers
102 views

Inverse function — need help

I'm a senior software developer but my math lessons are a bit rusty. I know the name of what I want, but not anymore how to compute it ;) I've found (by myself with Grapher.app) a simple easing ...
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2answers
5k views

Matrix is singular to working precision

I have a problem while evaluating inverse using inv in MATLAB. My matrix looks like this: ...
4
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1answer
700 views

Inverse of a diagonal matrix plus a Kronecker product?

Given two matrices $X$ and $Y$, it's easy to take the inverse of their Kronecker product: $(X\otimes Y)^{-1} = X^{-1}\otimes Y^{-1}$ Now, suppose we have some diagonal matrix $\Lambda$ (or more ...
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2answers
58 views

The inverse of $A+O(N^{-1})$

Assume $A$ is invertible and I want to calculate $(A+O(N^{-1}))^{-1}$ I want to know if there exist any formula for it? $O(N^{-1})$ is the big $O$ notation. That is the inverse of an invertible ...
3
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4answers
133 views

How to show $AB^{-1}A=A$

Let $$A^{n \times n}=\begin{pmatrix} a & b &b & \dots & b \\ b & a &b & \dots & b \\ b & b & a & \dots & b \\ \vdots & \vdots & \vdots & ...
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0answers
208 views

quadratic form of trace_inverse of symmetric positive definite matrix

I have the following problem: I need to implement a program that doesn't accept the matrix quadratic form $B^T\times B$ but it accepts the scalar quadratic form instead. Actually I need to find a ...
6
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504 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 $$(I+BA)^{-1} = A^{-1}(I+AB)^{-1}A$$