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.
4,339
questions
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Efficient way to compute the inverse of structured matrix
Let $Z\in R^{n\times p} ($p<n$), X=Z_{-ij}\in R^{n\times (p-2)}$ with the $i$ and $j$-th columns in $Z$ omitted. Given the eigen-decomposition of $ZZ^T=UDU^T$, is there any way to efficiently (...
- 337
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1
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20
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Invertibility of Block Matrix Partial Transpose
Let $$M = \left[\begin{matrix}
M_{1,1} & M_{1,2} & \cdots & M_{1,n}\\
M_{2,1} & M_{2,2} & \cdots & M_{2,n}\\
\vdots & \vdots & \ddots & \vdots\\
M_{n,1} & M_{n,...
- 145
3
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1
answer
33
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Finding a matrix and its inverse in terms of another one.
I have a symmetric matrix $A$. I wanted to modify its first row (thus it is first column). Let's call the new row $c^T$. How do I write the modified matrix $B$ in terms of $A$ and $c$. What is the ...
- 31
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31
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Polynomial evaluated at nilpotent matrix is invertible [duplicate]
Let $A$ be a nilpotent matrix, and let $p$ be a polynomial that satisfies $p(0) \neq 0$. How to prove that $p(A)$ is invertible?
- 1
-3
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52
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Can we find the inverse of matrix that are not a square?
For example, I was given a question where I had Matrix $A_{3×2}$ and was asked to find $U$.
$B$ (RREF of matrix $A$) = $UA$
Despite matrix $A$ not being a square my textbook did an augmented matrix ...
- 9
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0
answers
26
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Let $A$ and $B$ be $n \times n$ matrices. If $AB = I$, then $BA = I$. [duplicate]
Here is Prob. 15, Sec. 2.2, in the book Linear Algebra With Applications by Steven J. Leon and Lisette de Pillis, tenth edition:
Let $A$ and $B$ be $n \times n$ matrices. Prove that if $AB = I$, then ...
- 25.3k
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34
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Does the Sherman-Morrison formula apply with "rank-one" block matrices?
Consider the matrix
$$ B = A + \begin{bmatrix} R_{1} & R_{2} \\ R_{1} & R_{2} \end{bmatrix} = A + \begin{bmatrix} I \\ I\end{bmatrix} \begin{bmatrix} R_{1} & R_{2} \end{bmatrix} $$
where $...
- 1,129
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14
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Find the inverse of a map that changes 2D coordinates [closed]
Let $\boldsymbol{\Psi}: (x,y)\ \mapsto (\tilde x, \tilde y)
$
be defined as
$$
\boldsymbol{\Psi}(x,y)=
\begin{pmatrix}
\tilde x\\
\tilde y
\end{pmatrix}
:=
\begin{pmatrix}
ax+b\left(\frac{x}{\sqrt{x^2+...
- 1
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1
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43
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Procedure for finding the inverse of a $3 \times 3$ matrix — intuition?
I understand matrices in the light of linear transformations, and I know that the inverse of a matrix ($A^{-1}$) applied after the transformation $A$ essentially gets you back to where you started — ...
- 236
-2
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1
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50
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If $I-ABC$ is invertible then so is $I-BCA$ [closed]
$I-ABC$ is an invertible matrix. Prove that $I-BCA$ is also invertible.
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Show $(XX')^{-1}= PD^{-2}P'$, where $X = PDQ'$ is a thin SVD
$X$ is $n \times p$ dimension matrix.
$X = PDQ'$ is a thin SVD, where $P$ is $n \times r$, $D$ is $r \times r$, and $Q$ is $p \times r$.
Here is what I tried:
$$(XX')^{-1} = (PDQ'QDP')^{-1} = (PD^2P')^...
- 303
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1
answer
41
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How can I find a right-inverse of a fat matrix over $\Bbb F_2$?
Suppose I have a full rank fat binary matrix $A \in \{0,1\}^{m\times n}$ with $m<n$. How can I find a matrix $A^{\dagger}$ such that $AA^{\dagger}=I_{m}$?
The traditional method used for matrices ...
- 162
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1
answer
56
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Finding an inverse for Clausen Functions
General
Question: What does an inverse function for the Clausen Function look like?
Background: I'm doing some math with infinite series of trigonometric functions. I keep coming back to ...
- 996
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67
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Why is inverse multiplication shown as commutative?
Can someone explain why this is shown as commutative when matrix multiplication is not?
Is this a specific case with inverses?
"A square $(n\times n)$ matrix $A$ is said to have an inverse $A^{-1}...
- 9
1
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1
answer
42
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The null space of an invertible matrix
The statement reads as follows "If 𝐴 is invertible, there are no special solutions in the null space."
According to me, its actually true because if the matrix is invertible, it means that ...
- 25
-3
votes
1
answer
58
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Why only numbers coprime with n in A mod n have modular inverse? [duplicate]
Why do only the numbers coprime to n (numbers that share no prime factors with n) have a modular inverse (mod n)? Can anyone intuitively explain it?
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60
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How can I reformulate the following determinant?
I want to reformulate the following determinant:
$$ \left| \left(A^TB^{-1}A+ \frac{1}{c}C^{-1} \right)^{-1} \right|^{1/2} $$
where $c $ is a real scalar, $A$ is an ($n\times k$) matrix, $B$ is an ($n\...
- 67
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1
answer
35
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Trying to come up with matrix inverses algebraically — how does one do this?
I've been trying to find the inverse of a matrix through this sort of roundabout manner but to no avail. Sure, I could just use the standard way, but I figured working through this particular process ...
- 113
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1
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28
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$f(x, y, z):=\left(x+y^2+z^2, x-y+z, 2 x+y-z\right)$ local and global inverse, calculating $Df^{-1}$
We consider the mapping $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ given by
$$
f(x, y, z):=\left(x+y^2+z^2, x-y+z, 2 x+y-z\right) .
$$
a) Show that $f$ has a local inverse around the point $(1,-1,2)$.
...
- 101
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4
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100
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How to prove that if a $3 \times 3$ matrix has two equal rows, it has no inverse?
In my maths classes in school we have said that if a matrix has two equal rows then it has no inverse.
I can see this by calculating that the determinant of any $3 \times 3$ matrix with two equal rows ...
- 41
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votes
0
answers
10
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Tightening error bounds backwards through multiplication operation
Assume I have three scalar random variables $a,b,c \in \mathbb{R}$ related to each other through a multiplication operation:
$$ab=c$$
Let us further assume that we have three real-valued bounds for ...
- 917
2
votes
1
answer
49
views
Need help with finding the inverse of a matrix using row reduction
I've been trying to find the inverse of the matrix
\begin{pmatrix}1&0&-2&1&0&0\\ 3&1&-2&0&1&0\\ -5&-1&9&0&0&1\end{pmatrix}
using row ...
- 177
-1
votes
0
answers
23
views
Property of Invertibility for integer matrix
Prove that for any integer matrix $A \in M_{4}(\mathbb{Z})$, matrix $A^{4} + I$ is invertible or zero. Is it true for a matrix $A \in M_{4}(\mathbb{R})$
- 1
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0
answers
75
views
Does the grand sum of the inverse of a matrix decrease if we increase one element?
Let $\bf A$ be a symmetric and positive definite matrix. Assume that we modify $\bf A$ so that some element $a_{ij}$ with $i \neq j$ becomes $a_{ij} + \varepsilon$, where $\varepsilon > 0$ (note ...
- 21
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74
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Prove that $B = A^2 - 2A + 2I$ is an invertible matrix [duplicate]
If $A$ is a matrix such that $A^3 = 2I$, prove that $B = A^2 - 2A + 2I$ is an invertible matrix.
I tried to use $A^3 = 2I$ and turn it to $(A-I)(A^2 + A + I) = I$ but it didn't work. also I tried to ...
-1
votes
2
answers
84
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Why is $(X^{T}X+cI)^{-1} = (X^TX)^{-1}$ for ridge regression?
Why is $(X^{T}X+cI)^{-1} = (X^TX)^{-1}$? ($c$ is a scalar.)
My teacher in the lecture today used this fact when showing that the projection matrix of the ridge regression is idempotent. I was ...
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votes
1
answer
47
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$y=\ln \cos x$ inverse function flawed
The Domain of $f(x)$ is $(0,1)$.
When finding the domain of $f(\ln(x))$
We can say: $0<\ln x<1$ and apply the inverse $\ln$ function $\exp$.
We get $e^0 < e^{\ln(x)} < e^1$ which leads to $...
-2
votes
1
answer
28
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Invertability of sum of matrices [closed]
Consider two different n$\times$n matrix A and B generated by vector outer product. Therefore, both A and B are symmetric positive semidefinite but not invertible. In this case, is there a condition (...
- 1
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vote
1
answer
84
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Average of the inverse distance between two points in a uniform ball
I'd like to compute the average of the inverse distance between two different points $x=(x_1,x_2)$ and $y=(y_1,y_2)$ in a ball $B(0,R)$. In particular, I'd like to calculate the following integral (...
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0
answers
18
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Let $A$ be a nonsingular M-Matrix. Then $A$ has only eigenvalues with positive real parts.
Let $A$ be a nonsingular M-Matrix. How to show that the eigenvalues of $A$ have only positive real parts ?
Here a nonsingular M-Matrix $Q = (q_{ij})$ is a nonsingular Matrix such that $q_{ij} \leq 0 \:...
- 737
2
votes
1
answer
65
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The inverse of matrix $A^TBA$
Given tall matrix $A \in \mathbb{R}^{m \times n}$ (where $m > n$) of full column-rank and non-singular matrix $B \in \mathbb{R}^{m\times m}$, how to compute the following matrix inverse?
$$ \left( ...
- 158
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1
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81
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Compute $f^{∗∗} : \mathcal{PP} \emptyset → \mathcal{PPP} \emptyset$ for the unique function $f : ∅ → P∅$.
Question
Given an arbitrary function $f:X → Y$ , consider the new function $f^∗: \mathcal PY → \mathcal PX$ which is defined by the mapping $B → f^{−1}[B]$. Compute $f^{∗∗} : \mathcal{PP}∅ → \mathcal{...
- 3,552
1
vote
2
answers
93
views
How to prove that $B$ is invertible and to find $\det A$
I try to solve a question but can't find a way.
Given $A,B$ are $4\times4$ matrices and
$$
A-A^2B-2I = 0
\quad\text{and}\quad
3BA^2+A^2-3A = 0
$$
I need to prove that $A,B$ are invertible and to ...
- 95
1
vote
1
answer
67
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Finding the inverse of a matrix in terms of other matrices [closed]
Suppose $X$ and $Y$ are invertible square matrices of the same dimension. If $M=I-XY$ and $N=I-YX$ (and assuming $M$ is invertible), how do we find $N^{-1}$ in terms of $M$ and $X$?
- 41
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Bounding the inner product of a vector of correlations
Suppose $X$ is a Gaussian random variable and $Y$ is a Gaussian random vector of length $n$ (they are also jointly Gaussian).
Let $z$ be a vector with entries $z_i = \frac{\mathrm{Cov}[X, Y_i]}{\sqrt{\...
- 65
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votes
1
answer
142
views
Find the inverse of the function $f(x) =\frac{\ln(x)}{1-x^2}$
I'm trying to prove a certain integral converges, and for that I need to find the inverse of $f(x) =\dfrac{\ln(x)}{1-x^2}$. I've gotten this far:
$x = \dfrac{\ln(y)}{1-y^2} \rightarrow e^x = e^{\left( ...
- 344
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votes
0
answers
100
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Solving a quadratic program with one equality constraint
Let $d,n$ be positive integers, $(\mu_i)_{1\leq i \leq n}$ and $(\eta_i)_{1\leq i \leq n}$ some scalars, and define
$$ A := \displaystyle \sum_{i=1}^{n} \mu_i y_i y_i^T, \qquad c := \displaystyle \...
- 115
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1
answer
28
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Top-left element of inverse of real symmetric matrix
Let
$$
M =\begin{pmatrix} a & v^\top \\ v & B \end{pmatrix},
$$
with $a \in \mathbb{R}$ and $a \neq 0$, $v \in \mathbb{R}^{n-1}$, $B \in \mathbb{R}^{n-1 \times n-1}$ invertible and symmetric.
...
- 1,223
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14
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Inverse of a Set-Valued Transformation
How is the inverse of a set-valued transformation defined?
For a function $f$, we know that the inverse exists if and only if $f$ is bijective. However, I have found in a book that the inverse of
$$S:[...
- 520
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votes
2
answers
70
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Invertibility and inverse of a linear operator
Being an undergraduate student of functional analysis, I am trying to show that the following linear operator is invertible and find its inverse.
Let $C[0,1]$ be the space of real-valued continuous ...
- 11
1
vote
1
answer
32
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Involutions and matrices.
Let $a$,$b$, $c$ be members of a field. (?)
I noticed that $$f:\left(-\infty, \frac{a}{c}\right)\cup \left(\frac{a}{c}, \infty \right)\to \left(-\infty, \frac{a}{c}\right)\cup \left(\frac{a}{c}, \...
- 1,062
1
vote
1
answer
42
views
How many invertible affine functions $\mathbb{Z_2}\rightarrow \mathbb{Z_2}$
I need to know if there are any non affine invertible functions $\mathbb{Z_2}\rightarrow \mathbb{Z_2}$ and the tip given in the book is to take in consideration how many invertible affine functions ...
- 881
-1
votes
1
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63
views
Equivalent of $(A+B)^{-1}$. [duplicate]
I am working on inverse matrices and I have raised a problem about the inverse of the sum of two matrix. I have searched for quite a while but still could not find an answer. Does anyone know what is $...
0
votes
1
answer
40
views
How to obtain this simplification
I was studying a linear algebra problem and I encountered this step in the solution :
\begin{aligned}
& \mathbf{1}^T\left(\mathbf{1 1}^T+I \sigma_v^2\right)^{-1} y =\left(\mathbf{1}^T \mathbf{1}+\...
- 87
0
votes
2
answers
75
views
Inverse of difference operators
Given some difference operator of e.g. the form
$$\Delta [f(x)] = f(x+1)-f(x),$$how would I go about finding its inverse $\Delta^{-1}$, so some type of "discrete antiderivative" so that $$\...
- 33
0
votes
0
answers
32
views
Finding a generalized inverse (approximation) for multi-factorials given $k$
I am trying to find the inverse of:
$$f(n) = \prod_{x=1}^{n}x^{0^{\left|\left(\operatorname{mod}\left(x,k\right)-\operatorname{mod}\left(n,k\right)\right)\right|}}$$
for $k, n \in \mathbb{Z}^+$ (as ...
- 1
2
votes
0
answers
30
views
Matrix expression of the scalar product between two vectors
My question is in regarding the adjugate matrix in the expression my teacher gave me of the scalar product: $$ \vec{x}\cdot \vec{y}=x^+ G y$$ being $x^+=\overline{x}^T$ the traspose of the conjugate, ...
- 91
0
votes
0
answers
45
views
I am supposed to check if $ab+a+b$ is a group. [duplicate]
Is $(R, *)$ with $* : R \times R \to R$ by $(a,b) \mapsto ab + a + b$ a group?
I've shown that it's associative and that $0$ is the neutral element, but couldn't find an inverse element.
Is there no ...
- 21
0
votes
0
answers
14
views
Derivative of determinant of matrix product of matrix and Hadamard product of 2 matrices
My problem is how to compute the partial derivative of $
f(
\boldsymbol{\Lambda}_{\mathsf{h}},
\boldsymbol{\Lambda}_{\mathsf{x}},
\boldsymbol{\Lambda}_{\mathsf{z}}
)
$ with respect to $
\boldsymbol{...
- 1
0
votes
1
answer
51
views
why is a diagonal element of a matrix greater than its determinant divided by the complementary principal minor?
In this paper by Fiedler (1964) : "RELATIONS BETWEEN THE DIAGONAL ELEMENTS OF TWO MUTUALLY INVERSE POSITIVE DEFINITE MATRICES",
it is written in the proof of Theorem (3.2) on page 6 that $a_{...
- 51