All Questions
Tagged with matrices matrix-inverse
83 questions
2
votes
1
answer
231
views
Trace inverse of random PSD matrix?
Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. I am interested in the expectation of $$E_{A}(\mathrm{Tr}( (A^T A + \lambda \...
- 25
2
votes
2
answers
236
views
When is $(I - X)^{-\top} \circ X = 0$?
I am currently looking at the following expression $(I - X)^{-\top} \circ X = 0$, where $\circ$ is the Hadamard product, $\top$ is the transpose, $I$ is the identity, and $X$ is non-negative and ...
- 59
2
votes
1
answer
841
views
Matrix derivative w.r.t. a general inverse form: $(A^TA)^{-1/2}D(A^TA)^{-1/2}$
I want to find derivative of matrix $(A^TA)^{-1/2}D(A^TA)^{-1/2}$ w.r.t. $A_{ij}$ where D is a diagonal matrix. Alternatively, it is okay too to have
$$\frac{\partial}{\partial A_{ij}} a^T(A^TA)^{-1/2}...
- 519
2
votes
1
answer
196
views
Closed form for the inverse of a special transition matrix
Let $b\in\mathbf{R}^n$ be fixed, and let $A\in\mathcal{M}_{n\times n}(\mathbf{R})$ be given but otherwise arbitrary. Let $a_1,\ldots,a_n$ denote the coefficients of the characteristic polynomial of $A$...
- 309
2
votes
1
answer
841
views
Complexity of pseudoinverse of a low-rank matrix
I have a large matrix $\mathbf{A}\in \mathbb{C}^{m\times n}$ with very low rank $r$, I find that the general complexity of finding its pseudoinverse is $\mathcal{O}(\max(m,n)^3)$, this is too high. ...
- 121
2
votes
0
answers
146
views
What are the name and inverse of an interesting integer matrix?
It is practicable to compute the matrix inverses
\begin{align*}
\begin{pmatrix}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 2 & 2^2 \\
\end{pmatrix}^{-1}
&=\begin{pmatrix}
1 & 0 &...
- 1,091
2
votes
0
answers
137
views
Decompose a rational matrix as an integer matrix and an inverse of integer matrix
Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
- 823
2
votes
0
answers
300
views
Bound for the inverse of a summation of rank-1 matrices
Given vectors $x_1,\dots,x_T \in R^d$ satisfying $\|x_i\|_2 = 1$, define $A_0 = I$ and $A_t = I + \sum_{i=1}^tx_ix_i^\top$ for $t \geq 1$. We are interested in the following quantity:
\begin{align}
S_{...
- 21
2
votes
0
answers
529
views
Inverting a matrix with entries equal to positive or negative infinity
I would like to define an inverse on matrices whose entries may be positive or negative infinity.
To formulate my problem precisely, suppose that I have a matrix $A$ and another matrix $B$. How do I ...
- 45
2
votes
0
answers
2k
views
Update the inverse of sum of two symmetric matrices
There are two invertible symmetric matrices A and B, of which B is a block diagonal. A and B have the same dimensions. I need to iteratively calculate the inverse of M = s * A + B, where s is a ...
- 21
2
votes
0
answers
355
views
Fast inversion of a special kind of matrices - approximations are ok
Suppose I have a stochastic matrix $M$ (with thousands or millions of stochastic column vectors), which I split into two matrices: $D$ containing only the diagonal entries of $M$, and $R$ containing ...
1
vote
2
answers
732
views
Inverse of a matrix expression
Let
$$X_i = \left(I - P\left(I - t_it_i^T\right)\right)^{-1}$$
where $P$ is an $N\times N$ matrix and $t_i$ is a vector of $N$ elements.
Is there a way to simplify this expression in order to ...
- 151
1
vote
1
answer
499
views
A closed-form expression for the inverse of a block-matrix
Let $\bf A$ be an $n \times n$ non-singular matrix over $\mathbb{F}$.
Let $x$ be a non-zero element of $\mathbb{F}$.
Suppose that ${\bf 1}_{n}$ is a symbol for the all-one vector of length $n$ ...
- 211
1
vote
1
answer
252
views
Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible
$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix:
$$
M:= \begin{bmatrix}
\cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\
\cPr{X=...
- 151
1
vote
2
answers
143
views
Controllability Gramian asymptotics for small times
Set-up. Consider the following linear controlled system
$$
\dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1)
$$
where $y$ is the state of the system, $y(t) \in R ^n$, $A \in R ...
- 724
1
vote
1
answer
2k
views
Largest element in inverse of a positive definite symmetric matrix [closed]
If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
- 15
1
vote
1
answer
171
views
How to calculate $(A^{-1})_{ii}$ for an invertible hyperhermitian quaternionic matrix $A$?
The article
Alesker, S. (2003). Quaternionic Monge-Ampere equations. The Journal of Geometric Analysis, 13(2), 205-238.
has the following CLAIM:
Claim. Let $A$ be an invertible hyperhermitian ...
- 155
1
vote
1
answer
18k
views
Derivative of log determinant and inverse
I have a matrix $\Sigma$ with element $(i,j)$
$$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$
The matrix is positive definite and symmetric (it is a covariance matrix).
Now I need to evaluate
$$\frac{\...
- 29
1
vote
1
answer
345
views
Solvability of $A X B=C$ with $X=X^\mathrm{T}$
I am studying symmetric solutions to the complex matrix equation
\begin{equation}
A X B=C,
\end{equation}
where $A$, $B$, and $C$ are $m\times n$, $n \times k$, and $m \times k$ complex matrices, ...
- 61
1
vote
1
answer
245
views
Low-complexity method for sub-matrix inversion
Assume that $\mathbf{A}$ be an $N\times N$ matrix. We know that the complexity of the computation of matrix inversion is $\mathcal{O}(N^3)$. Let define $\mathbf{D}=\mathbf{A}^{-1}$. Now, assume that $\...
- 287
1
vote
1
answer
1k
views
Approximate $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$ by Taylor expansion [closed]
Suppose we have a complex matrix $\mathbf{M}$. Let $\mathbf{M}^+=(\mathbf{M}^*\mathbf{M})^{-1}\mathbf{M}^*$ be the pseudo-inverse of $\mathbf{M}$, where $^*$ denotes the conjugate transpose. Let $\...
- 233
1
vote
2
answers
1k
views
Is there an easy way to find the inverse of a symmetric block matrix with off-diagonal rank-one matrices?
The symmetric matrix I need to invert is of the following form:
\begin{align}
J_e=\left(\begin{matrix}-I&B_{11}&B_{12}&...&...&B_{1(N-1)}\\
B_{11}&-I&B_{22}&....
- 41
1
vote
0
answers
95
views
Algorithm for efficiently calculating $(A+\sum_{i=1}^n B_i)^{-1}$ where $A^{-1}\in\mathbb S^n_+$ is known and $B_i$ are sparse matrices
Let $A\in\mathbb R^{n\times n}$ be a symmetric positive-definite matrix and $A^{-1}$ is already known. Now I want to compute the matrix $(A+\sum_{i=1}^n B_i)^{-1}$ where each $B_i$ is a sparse ...
- 601
1
vote
0
answers
111
views
Angle inequality for inverse of PD diagonal matrix
I have two diagonal, positive definite matrices $A$ and $B$, and I have that the angle between $Ax$ and $x$ is smaller than that of $Bx$ and $x$:
$$\frac{x^T A x}{\|Ax\|_2} \geq \frac{x^T B x}{\|Bx\|...
- 269
1
vote
0
answers
319
views
Derivative of complex matrix pseudo inverse with respect to real and imaginary components
I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$.
I am interested in evaluating the ...
1
vote
0
answers
235
views
positiveness of the inverse solution to Sylvester equation
I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form:
$$
\mathbf{M} = \begin{vmatrix}
\mathbf{A} & \mathbf{b} \\\
\...
- 111
0
votes
1
answer
127
views
Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix,$A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$
This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$.
As known, ...
- 27
0
votes
1
answer
127
views
update rule for the inverse after a rank-1 update plus scaled identity
Is there an update rule for $$\left(\tilde{X}^T\tilde{X}+\alpha\cdot I\right)^{-1}$$ with $\tilde{X}=[X\;\; a]$ as a function of $A\triangleq (X^TX)^{-1}$, $X$ and $a$?
I know that when $\alpha=0$ we ...
- 503
0
votes
1
answer
138
views
Does there exist a function $f(X)$ with the following gradient $\mathrm{Tr}[(I-X)^{-1}]\cdot g(X)$?
Let $f: \mathbb{R}^{n\times n} \to \mathbb{R}$ be a function that receives a square matrix and spits out a scalar. Does there exist a function $f$ such that the gradient $\nabla_Xf(X) = \mathrm{Tr}[(...
- 59
0
votes
1
answer
2k
views
How to calculate the inverse of the sum of kronecker products with the identity matrix
How to calculate $G^{-1}$ efficiently when $G$ is a large matrix knowing that:
\begin{eqnarray}
G=I⊗A + A⊗I
\end{eqnarray}
Or since i'm using $G^{-1}$ to multiply by some other matrix, how to find $...
- 21
0
votes
0
answers
35
views
What is the impact of individual estimate on each other in matrix inversion?
I am looking to understand the impact of each estimate on each other in matrix inversion.
Lets say I have a vector $A = \left[a_1, a_2 \right]^T$ of size $2 \times 1$ and $a_1$ and $a_2$ are related ...
- 1
0
votes
1
answer
155
views
Product of matrices equal identity
I need to solve the following equation for the matrix $P \in\mathbb{R}^{r\times d}:$
$$
((PAP^\top)^{-1} P S P^\top (PAP^\top)^{-1})^2 = I_r,
$$
where $S$ is a symmetric $d\times d$ matrix, $A$ is a ...
- 245
-2
votes
1
answer
262
views
Proving 2 matrices have the same trace [closed]
I found a problem in my textbook and I have tried solving it, but I had no succes. The problem is:
Let $A$ and $B$ be $n \times n$ matrices with complex number entries. Given that $AB−BA$ is ...
- 331