Linked Questions

1
vote
0answers
123 views

How to inverse $(I + \alpha M)$ for all $\alpha$ [duplicate]

I am looking for a way to solve the following equation: $$(I + \alpha M)X=F$$ $\forall \alpha \in R$ (the real domain), with $M$ a square complex matrix without any particular properties, $I$ the ...
10
votes
3answers
13k 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 ...
6
votes
2answers
3k views

Inverse of symmetric matrix plus identity matrix

Consider the symmetric, positive definite matrix $\mathbf{A}$. I'd like to find a general form for $$(\mathbf{I} + \mathbf{A})^{-1}$$ that only involves $\mathbf{A}^{-1}$, i.e., no other inverse ...
8
votes
2answers
2k views

Inverse of the sum of a symmetric and diagonal matrices

I have two matrices $A$ and $B$ with quite a few notable properties. They are both square. They are both symmetric. They are the same size. $A$ has $1$'s along the diagonal and real numbers in $(0 ...
37
votes
1answer
1k views

A System of Matrix Equations (2 Riccati, 1 Lyapunov)

Setup: Let $\gamma \in(0,1)$, ${\bf F},{\bf Q} \in \mathbb R^{n\times n}$, ${\bf H}\in \mathbb R^{n\times r}$, and ${\bf R}\in \mathbb R^{r\times r}$ be given and suppose that ${\bf P}$,${\bf W}$,${\...
4
votes
1answer
3k views

Inverse of matrix sum of identity and outer product

So before we begin, I already know the answer. I'm just having difficulty figuring out the steps for finding it. Given $u,v \in \mathbb{R}^{n}$, I want to show that $$(I+uv^{T})^{-1}= I - \frac{uv^{T}...
3
votes
3answers
1k views

Least Squares with Euclidean ($ {L}_{2} $) Norm Constraint

Suppose I have set of samples $(x_i,y_i), 1 \leq i \leq n$. I am interested in solving the following optimization problem: $$ \min \sum_{i=1}^n (y_i-a^\top x_i)^2, \quad \text{s.t } \|a\|_{2} = 1. $$ ...
5
votes
1answer
654 views

Is there a way to solve explicitly the following functional equation?

I want to find an unknown function (actually CDF) $F(p)$ which solves $1 - \lambda F(\frac{q_L}{q_H}p) - (1-\lambda)F(p-[q_H-q_L]) - \frac{K}{p-c_H} = 0$, where $0<\lambda<1$, $q_H > q_L &...
2
votes
2answers
375 views

Proof of matrix inverse

Prove that $(A+uv^T)^{-1}=A^{-1}-\frac{1}{1+v^TA^{-1}u}A^{-1}uv^TA^{-1}$ Can someone give a hint how to show it.I am not getting from where to start.
2
votes
1answer
938 views

Inverse of sum of two marices, one being diagonal and other unitary.

$C = A+D$, $A$ being square matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$ Edit 2: (important edit) Iam interested in this question, ...
1
vote
2answers
590 views

Inverse of a sum of PSD matrices

I was wondering if anyone knew any techniques to convert the following: $ (A+B+C+..)^{-1} $ where $A,B,C...$ are positive semi-definite (PSD) matrices into a sum of some other function: $ f(A)+f(B)+...
0
votes
2answers
536 views

Approximating the inverse of a perturbed matrix

Consider a matrix $A$ which we subject to a small perturbation $\partial A$. If $\partial A$ is small, then we have $(A + \partial A)^{-1} \approx A^{-1} - A^{-1} \partial A A^{-1}$ I came across ...
1
vote
1answer
489 views

How to convert $(A+\lambda E)^{-1}$?

Here is one of the most famous equation called Sherman–Morrison formula (1951) when we want to get an inverse matrix. $$(A+vw^{\text{T}})^{-1}=A^{-1}-\cfrac{A^{-1}vw^{\text{T}}A^{-1}}{1+{w}^{\text{T}}...
1
vote
0answers
584 views

Inverse of sum of 3 matrices

I need a way to compute the inverse of the sum of three matrices: $(A + BB^T + \beta I)^{-1} $ where $I$ is identity and $\beta$ is a constant. I am not very familiar with linear algebra, but a ...
4
votes
3answers
76 views

Find two $2\times2$ real matrix $A$ and $B$ such that $A$, $B$ , $A+B$ are all invertible with $(A+B)^{-1}=A^{-1}+B^{-1}$

Find two $2\times2$ real matrices $A$ and $B$ such that $A$, $B$ , $A+B$ are all invertible with $(A+B)^{-1}=A^{-1}+B^{-1}$ Tried to write the matrices as $$A=\pmatrix {a&b\\c&d},B=\pmatrix {...

15 30 50 per page