Linked Questions

-2
votes
0answers
37 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
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
39 views

Prove that matrices $B$ and $(I+B)^{-1}$ commute

From following derivation, depending on step, where to substitute $A=I$ one can obtain: $$ (I+B)^{-1}=I-B(I+B)^{-1} \\ (I+B)^{-1}=I-(I+B)^{-1}B $$ From which follows that matrices $B$ and $(I+B)^{-...
1
vote
1answer
32 views

Matrix function that gives a scalar

I have the following function: $$f(z) = z\vec{b}^T[I-zA]^{-1}\vec{1},$$ where $z$ is a complex scalar with $Re(z)<0$ (for simplicity, WLOG, we can take $z$ to be real), $b$ is a vector, $1$ is a ...
1
vote
0answers
27 views

Reduced complexity of matrix inversion of sum of rank 1 matrices: $M^{-1} = \left( I + \sum \limits_{i=1}^{K} \alpha_i A_i \right)^{-1} $?

How to reduce the complexity of matrix inversion of sum of rank 1 matrices, not only arithmetic but also run time? $M^{-1} = \left( I + \sum \limits_{i=1}^{K} \alpha_i A_i \right)^{-1} $ where $A_i ...
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
0answers
134 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 ...
0
votes
0answers
26 views

Convexity of a function over matrices [duplicate]

How to show that $f(X)=tr(X^{-1})$ is convex over the set of symmetric real positive definite matrices by showing that $g(t)=f(X+tV)$ is convex for any such $X$ and $V$? Clearly we need to show that $$...
0
votes
2answers
591 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 ...
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 {...
0
votes
0answers
16 views

Simplification of $x = \left(A + \sum_i \alpha_i B_i\right)^{-1} A \ y$, $A \succ 0, A \in M_{n,n}(\mathbb{C})$, $\textrm{rank}(B_i) = 1$

I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition. Problem definition: Say $A \...
6
votes
2answers
4k 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 ...
-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 ...
1
vote
1answer
93 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
122 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}^{...

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