All Questions
Tagged with matrix-decomposition inverse
54
questions
0
votes
0
answers
28
views
Woodbury matrix identity with a minus sign
Is there a form of Woodbury matrix identity
$(A + UCV )^{-1} = A^{-1} - A^{-1}U (C^{-1} + VA^{-1}U )^{-1} VA^{-1}$
But with a minus sign? i.e. $(A - UCV )^{-1}$ It seems like I have to painfully ...
- 1,634
1
vote
1
answer
30
views
Any decomposition of inverse of nonnegative diagonal matrix times a PSD matrix plus lambda times Identity?
I generally have to solve the following system:
$$
(DA + \lambda I)^{-1} v
$$
where $D$ is a diagonal matrix with nonnegative entries, $A$ a symmetric, positive semi-definite (PSD) matrix, $I$ is the ...
- 133
0
votes
0
answers
42
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:
$...
- 72
3
votes
2
answers
138
views
Derivation of inverse matrix [duplicate]
My question is about how can reach to the formula $$\frac{dA^{-1}}{dt}=-A^{-1}\frac{dA}{dt}A^{-1}$$ when $A$ is a matrix. It is very similar to $$\frac{dx^{-1}}{dt}=-x^{-2}\frac{dx}{dt}$$And I know we ...
- 25.2k
2
votes
2
answers
90
views
Inverse of $3\times3$ block upper triangle matrix
How to find the inverse of $3\times 3$ block upper triangular matrix
$$X = \begin{bmatrix}
\mathbb{1} & \mathbb{B} & 0\\
0 & \mathbb{1} & \mathbb{B}\\
0 & 0 & \mathbb{1}
\end{...
- 151
0
votes
0
answers
49
views
Inverse of part of block matrix
I am trying to write a proof and, in order to do so, I have to simplify the following
$$ \left( E H B^{-1} \right)^b \bigg(\big(V-VH(B+HVH)^{-1}HV\big)^{bb}\bigg)^{-1}$$
where $b$ indicates the rows ...
- 70
0
votes
1
answer
41
views
How to obtain this simplification
I was studying a linear algebra problem and I encountered this step in the solution :
\begin{aligned}
& \mathbf{1}^T\left(\mathbf{1 1}^T+I \sigma_v^2\right)^{-1} y =\left(\mathbf{1}^T \mathbf{1}+\...
- 105
1
vote
0
answers
60
views
Best inverse / minimization solution of ill conditioned matrix and underdetermined system
I have an ill-conditioned matrix G representing Green's function estimates of a strain model for a rock mass that has been instrumented with optical fiber measuring strain. My problem is seemingly the ...
- 11
1
vote
0
answers
58
views
Compute $PSP^{-1} y$ without computing matrix inverse
Let $P$ (symmetric) and $S$ be two matrices of size $N \times N$, and $y$ be a vector of size $N \times 1$. I would like to compute $PSP^{-1} y$ without computing $P^{-1}$ because, in my case, $P$ is ...
- 338
2
votes
0
answers
128
views
Numerical stability of a matrix inversion problem
Suppose, we have a matrix X with dimension m and n.
Now lets say, we are interested to find the inverse of the covariance matrix: $X^T X$. The straightforward way to do it is to compute the matrix $X^...
- 287
1
vote
0
answers
99
views
Computing inverse elements of symbolic matrices with binary variables
I'm working with symmetric, symbolic matrices $A$ with real coefficients and linear binary variables like
$$
A = \begin{pmatrix} 0.5x_0 & 0.3x_1+0.002x_2 & 0 & 0 \\
0....
- 11
1
vote
0
answers
60
views
Decomposition of inverse of sum of block diagonal matrices
I have $K$ binary matrices $Z_k$ of dimension $n \times q_k$. These matrices are the kind you would get when performing one-hot encoding on a categorical variable, e.g. for $n = 7$ and a categorical ...
- 177
0
votes
1
answer
47
views
Question about inverse of a matrix.
Consider the block upper triangular matrix
$$A = \left[ \begin{matrix} A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} \right], $$
where $A\in\mathbb R^{n\times n}$ and $A_{11}\in\mathbb R^{k\times ...
- 744
1
vote
0
answers
49
views
How to find pseudoinverse Moore–Penrose matrix
I have $\begin{align}
A &= \begin{bmatrix}
2 \\
3 \\
\vdots \\
\sqrt{5n-1}
\end{bmatrix}
\end{align} $*$\...
- 31
0
votes
0
answers
36
views
$(I+vw^T)^{-1}=I -\dfrac{vw^T}{1+w^Tv}$ implies $w^Tv=vw^T$
Let $I$ be a $n$ x $n$ matrix, and $v$ and $w$ a nonzero $n$ x $1$ column matrix. Imagine $vw^T$ is nonsingular (therefore $(vw^T)^{-1}$ exist).
Here is my prove:
$(I+vw^T)^{-1}=I-\dfrac{vw^T}{1+w^Tv} ...
- 33