Linked Questions

1
vote
0answers
131 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 ...
-2
votes
0answers
36 views

How to find the inverse of sum of matrices [duplicate]

If I have matrix A and matrix B how do I find the inverse of (A+B), the sum of the matrices and A is invertible but B isn’t
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
2k 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
656 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
377 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
1k 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
598 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
583 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
491 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
592 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 ...
2
votes
1answer
212 views

Inverse or approximation to the inverse of a sum of block diagonal and diagonal matrix

I need to calculate $(A+B)^{-1}$, where $A$ and $B$ are two square, very sparse and very large. $A$ is block diagonal, real symmetric and positive definite, and I have access to $A^{-1}$ (which in ...
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 {...
1
vote
2answers
150 views

Inverting a special matrix

Consider matrices $A$ and $B$ of the forms below: $$A = \lambda \cdot I$$ $$B = \beta \cdot \pmatrix{ 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & \...
3
votes
1answer
232 views

Show that $A^T=A^{-1}$

Say I have a matrix $B$ that is skew symmetric ($B^T=-B$). I want to show that for $A=(I+B)(I-B)^{-1}$ that $A^T=A^{-1}$ is true. All we know is that $B$ is square and that $(I-B)$ is non singular. ...
0
votes
1answer
463 views

Concavity of trace of positive definite matrix

I have to show that $Tr((A^{-1} + B^{-1})^{-1})$ is a concave function, being A and B positive definite matrices. I cannot imagine how is this possible since we are computing the trace of a positive ...
0
votes
1answer
341 views

harmonic mean of covariance matrices.

After writing up some math, I ended up with a term like so: $\left(A^{-1} + B^{-1}\right)^{-1}$ where $A$ and $B$ are 2 covariance matrices. 1) Can I be sure that this expression is meaningful? (i....
1
vote
0answers
361 views

Pseudoinverse of the sum of matrices

Similarly to the question posted here Inverse of the sum of matrices but in case of non-square matrices. If I want to compute the pseudoinverse of (A+B) and matrices A,B pinv(A) is known is there a ...
4
votes
1answer
128 views

Is there a way to update the inverse of a sum of two matrices following a rescaling of one of them?

Suppose I have two matrices $A$ and $B$ (let's assume that both $A$ and $B$ are invertible, as is their sum), and a scalar $g$. I am interested in the matrix $$M^{-1} = (A + gB)^{-1}$$ I am aware ...
1
vote
3answers
118 views

Any lemma for $(A+A^{-1})^{-1}$?

I'm actually a little surprised since I wasn't able to find any nice property to compute $(A+A^{-1})^{-1}$ ... Anyone knows about a theoretical way to achieve this ? Like an specific inversion lemma?...
0
votes
1answer
61 views

Inverse of the $(A + iB)$

I have taken this question from Zhang, Fuzhen. Linear Algebra (Johns Hopkins Studies in the Mathematical Sciences) Assuming that all matrix inverses involved below exist, show that $(A + iB)^{...
1
vote
1answer
92 views

If $A^{-1}$ has been precomputed, is there an efficient way to compute: $(A+λI)^{-1}$

If $A^{-1}$ has been precomputed (or to be more precise: the Cholesky decomposition of A has been precomputed and cached), is there an efficient way to compute either $$C = (A+λI)^{-1}$$ or (more ...
1
vote
1answer
121 views

Symbolic inverse of a linear combination of two matrices

This question is very much related to Inverse of the sum of matrices . Let $d\ge 1$ be an integer and let $a\in(-1,1)$ and $b\in(-1,1)$ be real numbers. Let ${\bf C}:=\left( f(|i-j|) \right)_{i,j=1}^{...
0
votes
1answer
95 views

Representation of the inverse of an variance-covariance matrix $\hat{\Sigma}^{-1}$

Given $T$ observed vectors $x_i\in\mathbb{R}^N, i\in\{1,\ldots,T\}$. Define $\hat{\Sigma}$ as the corresponding empirical covariance-matrix of the Observations $X=\left(\begin{array}{c} x_1' \\ \...
4
votes
0answers
70 views

Computation of Matrix inversion

Suppose $\mathbf{A}\in \mathbb{R}^{n\times n}$ is non-singular and its inverse $\mathbf{A}^{-1}$ is known. We would like to compute $\mathbf{B}^{-1}$ where we assume that $\mathbf{B} = \mathbf{A}+\...
-1
votes
1answer
53 views

Real parts of diagonal elements of $(I-U)^{-1}(I+U)$ are zero?

I noticed numerically that real parts of diagonal elements of $(I-U)^{-1}(I+U)$ are zero (assuming $I-U$ is invertible), where $I$ is identity matrix and $U$ is a unitary matrix ($U^\dagger U=I$). A ...

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