39 questions linked to/from Inverse of the sum of matrices
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### 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 ...
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### 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: $$\min \sum_{i=1}^n (y_i-a^\top x_i)^2, \quad \text{s.t } \|a\|_{2} = 1.$$ ...
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### 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. ...
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### 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 ...
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### 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.
$C = A+D$, $A$ 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, ...
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 & \...