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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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31 votes
4 answers
7k views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
0 votes
3 answers
4k views

How to prove that $A$ is invertible if and only if $A^k$ is invertible?

Let $A$ be an $n \times n$ matrix. Show that $A$ is invertible if and only if any power $A^k$ (with $k\geqslant1$) of $A$ is invertible. I've been looking over the Theorem of Invertible Matrices but ...
0 votes
0 answers
43 views

Is $A \in R^{n,n}$ invertible, if there exists an $m \in \mathbb{N}$ with $A^m = I_n$? [duplicate]

Let $R$ be a commutative ring with 1 ($1 \ne 0$). Let $A \in R^{n,n}$ be a matrix and there exists an $m \in \mathbb{N}$ with $A^m = I_n$. The number $m$ is as small as possible to hold this property. ...
0 votes
2 answers
62 views

Is this special BLOCK upper triangular matrix diagonalizable?

Let $A$ be a block upper triangular matrix: $$A = \begin{bmatrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{bmatrix}$$ where $A_{1,1} ∈ {\mathbb{R}}^{p \times p}$, $A_{2,2} ∈ {\mathbb{R}}^{(q) \times (...
0 votes
0 answers
58 views

Inverse trigonometry , find x [closed]

What is the value of x ? There are many different ways to do . Is my method correct? If so can anyone help me to go further. After this part , i have no idea to go with !!
0 votes
0 answers
12 views

Change a matrix of base with respect a special base consisted of a kernel and a rowspace

I have got a matrix $B$ $m \times n$. I found a basis of its Kernel and of its rowspace so both together form a basis of $\Re^n$. I have seen in an article that you can now express a matrix $A$ $n \...
0 votes
0 answers
11 views

Inverse of a Block matrix formed by two column blocks which represent orthogonal complements

I have got a matrix $B$ $m \times n$ with rank $m$. I found then the basis of its kernel and of its rowspace. Both together form a basis of $\Re^n$. With this I make the change of basis matrix $P$ ...
0 votes
0 answers
20 views

Proving $(S \circ R)^{-1} = R^{-1} \circ S^{-1}$ [duplicate]

I need to prove the following statement from Velleman's How to Prove It (p. 176): Suppose $R$ is a relation from $A$ to $B$ and $S$ is a relation from $B$ to $C$. Prove that $(S \circ R)^{-1} = R^{-1}...
1 vote
1 answer
27 views

Inverse and Composition of Bisimulations

Exercise 63 of Rutten's The Method of Coalgebra: exercises in coinduction asks us to prove that "the collection of all bisimulation relations between two given stream systems is closed under (i) ...
0 votes
0 answers
72 views

Problem 46, section 2.5 on Introduction to Linear Algebra, 4th edition.

The question asks how does the identity $A(I+BA)=(I+AB)A$ connect the inverses of $I+BA$ and $I+AB$, and I am able to express $(I+BA)^{-1}$ as $A^{-1}(I+AB)^{-1}A$. However, I fail to see how this ...
1 vote
2 answers
956 views

How to calculate the inverse of sum of a Kronecker product and a diagonal matrix

I want to calculate the inverse of a matrix of the form $S = (A\otimes B+C)$, where $A$ and $B$ are symetric and invertible, $C$ is a diagonal matrix with positive elements. Basically if the ...
9 votes
3 answers
1k views

Inverse matrix of matrix (all rows equal) plus identity matrix

Let $A$ be a matrix where all rows are equal, for example, $$A=\left[\begin{array}{ccc} a_{1} & a_{2} & a_{3} \\ a_{1} & a_{2} & a_{3} \\ a_{1} & a_{2} & a_{3} \end{array}\...
0 votes
1 answer
25 views

Modified graph Laplacian, D + A

Consider an undirected, connected graph with positive edge weights $G$ with adjacency matrix $A$ and (diagonal) degree matrix $D$, and graph Laplacian given by $L = D - A$. $L$ is singular and is non-...
24 votes
3 answers
41k views

What is inverse of $I+A$?

Assume $A$ is a square invertible matrix and we have $A^{-1}$. If we know that $I+A$ is also invertible, do we have a close form for $(I+A)^{-1}$ in terms of $A^{-1}$ and $A$? Does it make it any ...
3 votes
2 answers
368 views

If $A^2=O$, then prove that $I+A$ is invertible and find $(I+A)^{-1}$

So I'm stuck on this Linear Algebra question. My first (naive) train of thought here was to go through with an implication that $A^2=O$ implies $A=O$. Then this got quickly debunked having read up on ...
0 votes
1 answer
89 views

Why is (I+A) invertible and what is its inverse? [duplicate]

Let A and B be n x n (square) matrices where A^2 = 0 and B^2 = 0. Let I be the identity matrix. (I+A) is invertible. Why? What is its inverse? Does A^2 = 0 imply that the matrix A is nilpotent, making ...
0 votes
1 answer
28 views

Inverse of the identity minus a block anti-diagonal matrix.

Let $A$ be an $m\times n$ matrix and $B$ be an $n\times m$ matrix. Denote by $C$ the square block matrix given by $$ C=\left[\begin{array}{cc} 0 & B\\ A & 0 \end{array}\right]. $$ I am looking ...
0 votes
0 answers
20 views

The inverse of a specific case of symmetric matrix (scalar product of d dimensions vectors)

The problem is the following: For $i \in [N]$, let $v_i$ be a $1 \times d$ vector and $b_i$ a scalar. Moreover, let $A$ be a $N \times N$ matrix, whose (i, j)-entry is: $$ a_{ij} = \begin{cases} \...
0 votes
0 answers
32 views

Inverse of a matrix sum and difference in terms of known inverses

I have been working on a problem related to the inverses of matrices and would appreciate any insights or solutions. The problem is as follows: Given two invertible matrices $A$ and $B$ with known ...
1 vote
0 answers
31 views

Inverse function / mapping considering vector multiplication by matrix

Inverse function / mapping considering vector multiplication by matrix also touches symetric encryption Consider, there's a simple matrix as a mapping from R3 ➝ R3 ...
12 votes
1 answer
741 views

Taking the inverse (not the reciprocal) of both sides of an inequality

This is something I'm having a hard time finding online, but say we know that $f(x) > g(x)$ (for all inputs $x > a_{0}$ for some $a_{0}$), then would it always be true that $f^{-1}(x) < g^{-1}...
1 vote
0 answers
122 views

Sum of entries of the inverse of a positive definite matrix $I_{\ell} + A_1A_1^T + A_2A_2^T$

Let $A_1$ be an $\ell \times n$ matrix and $A_2$ be an $\ell \times m$ matrix, where $\ell \ge n, m$. The entries of $A_1$ and $A_2$ are only $0$'s and $1$'s such that Each row has exactly one $1$, ...
1 vote
1 answer
198 views

How to inverse the laplace transform $\frac{1}{\cosh(5\sqrt{s})}$?

Let $X$ be a random variable with $ E[e^{-sX}]=$ $\frac{1}{\cosh(5\sqrt{s})} $ and density function $f$. How to give a formula for $f$?
1 vote
0 answers
42 views

Verification of a demonstration

I need to know if the proof I made for the following problem is correct. Problem: If C is a matrix of order $3 \times 3$ such that $\text{rank}(C) = 2$, then $\text{det}(C) = 0$ Proof: If it must be ...
1 vote
0 answers
86 views

How to calculate $\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$ [closed]

I wonder if there is an analytic solution for the following equation: $$\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$$ Here, $a$ is a constant. Would you please give an advice?
1 vote
1 answer
959 views

Is it true that $X(X'X)^{-1}X'-J/n$ is idempotent, where $J$ is an $n$ by $n$ matrix of ones?

$X$ is a full column rank $n$ by $p$ matrix with the first column a vector of ones. Now the I was trying to prove, from a different approach that the SSR/variance is Chi square but this means I have ...
0 votes
0 answers
40 views

Quadratic form of a matrix where non-standard decomposition is known

Let $1\le m<<n$ for integers $m,n$. I have two symmetric matrices of size $n\times n$, say $A$ and $B$. My goal here is to know a simple form, or an upper bound on the following quadratic form: $...
0 votes
1 answer
29 views

Is the product of a right-invertible, an invertible and a left-invertible matrix itself invertible?

Suppose $B \in \mathbb{R}^{(n,n)}$ is invertible and $A \in \mathbb{R}^{(n,m)}$ is left-invertible. Is $A^T B A$ invertible? I know that $A^T A$ is invertible. I've been trying to work it out using ...
0 votes
1 answer
63 views

How to find the matrix $A$ from $C_0=A \times B$ or $C_1=B \times A$, given the singluar matrix B.

Kindly help me in the following: Let $C_0=A \times B$ and $C_1=B \times A$, where $A$ is a full rank square matrix, $B$ is a non-zero singular square matrix. Entries in $A,B,C$ are from finite ...
6 votes
3 answers
7k views

How to Find Moore Penrose Inverse

I have a matrix: $$A= \begin{bmatrix} -1 & 0 & 1 & 2 \\ -1 & 1 & 0 & -1 \\ 0 & -1 & 1 & 3 \\ 0 & 1 & -1 & -3 \\ 1 & -1 &...
0 votes
0 answers
37 views

Is there any mathematical results stating when there are 0's in the inverse of a square matrix given 0's in the original matrix?

I am working with square invertible matrices. Denote the n-by-n matrix as $A \in \mathbb{R}^{n \times n}$. Say we know that there are some 0's in the matrix. For instance: $A_{ij} = 0$ for some $i,j$ ...
0 votes
0 answers
76 views

$A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f$ is a non constant polynomial.

The Actual Question $A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f=0$ has finitely many solutions. Thoughts I understand that $f$ is a non zero polynomial, and if it's not a constant ...
-2 votes
0 answers
29 views

Let the universal set be Z_55 = {0,1, ... , 54}. How many elements in Z_55 have an inverse in Z_55? [duplicate]

I am a little confused here, as I know I can use Eulers phi-function to calculate the amount of relative prime numbers up to 55, and that this then means there exists an inverse for each number mod 55....
1 vote
2 answers
2k views

Negative definite matrix and its inverse

$$ A= \begin{pmatrix} -62 & -158 \\ -158 & -398\end{pmatrix} $$ Is $A$ negative definite?(Is ${\displaystyle z^{\textsf {T}}Az} <0 $ for every non-zero column vector $z$ of $n$ real ...
3 votes
1 answer
1k views

Time complexity of inverting an $n \times n$ matrix which is the sum of a rank-$m$ matrix and a full-rank diagonal matrix

I want to know the time complexity of inverting $K$, where $K$ is an positive-definite $n\times n$ matrix: $$K=\Lambda+Q$$, where $\Lambda$ and $Q$ are both $n\times n$ matrix, $\Lambda$ is a full-...
0 votes
0 answers
27 views

Given $M, N$, how to find matrix $M'$ such that $M'N = NM$

Consider an arbitrary $p\times q$ matrix $N$, an arbitrary $q\times q$ matrix $M$. I do not know if $N$ is invertible - the solution is easy when it is. Is there a way to always find a $p\times p$ $M'$...
0 votes
1 answer
40 views

Invertibility of an integral matrix expression

Given $A:\mathbb R^n \rightarrow \mathbb R^{n\times n}$, $x\mapsto A(x)$ invertible for all $x$. In particular, it is known that $$A(x) = \frac{\partial}{\partial x} f(x)$$ with $f:\mathbb R^n \...
0 votes
0 answers
15 views

Inverse Relation for Allocating an Amount

This question came up in the context of project finance. It is my original question. Quick Version A project has upfront cost $X$, and ongoing cost $Z$. (Z is less than X). There are the same $N$ ...
1 vote
2 answers
115 views

Is $[\sin(\frac{1}{2})]^{-1}$ identical to $\frac{1}{\sin(\frac{1}{2})}$

I'm in Grade 12 Advanced Functions and having some trouble with understanding the difference between $\sin^{-1}(\frac{1}{2})$ and $(\sin(\frac{1}{2}))^{-1}$. I recognize that the former asks to find ...
0 votes
1 answer
2k views

On the production model ${\bf x} = {\bf C x} + {\bf d}$

Consider the production model ${\bf x} = {\bf C x} + {\bf d}$ for an economy with two sectors, where $$ {\bf C} = \begin{bmatrix} 0.0 & 0.5 \\ 0.6 & 0.2 \end{bmatrix}, \qquad {\bf d} = \...
0 votes
1 answer
53 views

Requirements for invertibility of $A B A^T$ in constrainted dynamics

What are the requirements for matrix $A$ (that isn't a square matrix), so that the matrix $A B A^T$ is invertible, given that $B$ is non-singular? Some details for the matrices: $B$ is the $n \times n$...
0 votes
1 answer
40 views

Inverse function theorem generalization

In inverse function theorem, it requires $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$. Can it be applied to the case that $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ where $n>m$. In other words, how ...
0 votes
1 answer
1k views

Matrix Inversion distribution

How do you distribute the inversion in $(A^TA+\lambda I)^{-1}A^Ty$ assuming $A$ is a $n \times n$ square invertible matrix, $y$ is a vector with the dimension of $n$, and $\lambda$ is a constant?
1 vote
1 answer
2k views

Inverse Function of sum of exponential function

What is the inverse function for $$y=a^x+b^x+...+z^x$$ where $a, b, .. , z$ are positive constant and $x>0$ Thanks in advance!
1 vote
3 answers
2k views

Invertibility of Gram Matrix - Proof

I'm an undergrad that just got introduced to ordinary least squares and am trying to understand why a Gram matrix is invertible only if the column vectors are linearly independent. The thread below (...
0 votes
1 answer
91 views

Why is $A^{-1}$ existing a necessary condition for $x$ to be unique in $Ax = b$? [duplicate]

Consider the equation $Ax = b$ where $A \in \mathbb{R}^{m \times n}$, and $x \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$. Say we know $A$ and $b$. I am wondering why, in order for us to uniquely ...
-2 votes
1 answer
78 views

Find the formula for the inverse transformation of $T\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}=(x+1)^2(2c-a-b)-(3x-2)(a+b)+x(5c-b)-3c$ [closed]

I was looking to the exercise for Linear Algebra from my course materials. Stumbled upon this question, and I have no idea how to start or solve this question. As I am totally new to the realm of ...
0 votes
0 answers
60 views

Block Matrix Inverse and Imaginary Number

I came here to ask some help regarding the following question. Let $M \in \mathbb R^{n \times n}$ be a "symmetric" and positive semidefinite matrix, and $M^{(n)}$ is obtained from $M$ by ...
5 votes
3 answers
5k views

Inverse of a lower triangular matrix

I got the following question to solve: Given the lower triangular matrix \begin{bmatrix} A_{11} & 0 \\ A_{21} & A_{22} \end{bmatrix} of size $n \times n$ (n is a power of 2) ...
1 vote
0 answers
14 views

Results of invertibility of a matrix involving the Szego kernel

In the context of reproducing kernel Hilbert spaces, the Szego kernel is the function $k(z_i,z_j)=\frac{1}{1-z_j\overline{z_j}}$. Given two sets of points $\{z_1,\ldots,z_n\},\,\{w_1,\ldots,w_n\}\...

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