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5 votes

Closed form formula for $(\mathbb I+R)^{-1}$ where $R$ is an orthogonal matrix?

If $-1$ is an eigenvalue of $R$, $\mathbb I + R$ is not invertible. Otherwise, let $P(x)= \det(x\mathbb I - R)$ be the characteristic polynomial of $R$. Note that $P(-1) = \det(-\mathbb I - R) \ne 0$....
Robert Israel's user avatar
4 votes
Accepted

Is $\text{Id}-T$ always invertible when $\lim_{n\to\infty} T^n = 0$?

Injective: Let us first prove that $I-T$ is injective. For this we check that the kernel is trivial. Let $v$ be in the kernel of $I-T$, i.e. $(I-T)v=0$, then $v=Iv=Tv$. However, this cannot be as ...
Severin Schraven's user avatar
2 votes

Iteratively finding matrix inverse from a given inverted matrix.

If $A$ is a good approximation of $B$, then $A^{-1}$ is not necessarily a good approximation of $B^{-1}$. As an example consider the case of $$A = \begin{bmatrix} 1 & 0 \\ 0 & 10^{-9} \end{...
Carl Christian's user avatar
2 votes
Accepted

Prove that the given function is invertible

This system \begin{align} u&=x+y+z^2\,,\\ v&=x-y+z\,,\\ w&=2x+y-z\, \end{align} can be solved: \begin{align} v+w&=3x\,,\tag{gives you $x$}\\ u+v&=2x+z^2+z=\tfrac32(v+w)+z^2+z\,,\...
Kurt G.'s user avatar
  • 15.3k
2 votes

$P=S^{-1}$ and $Q = S[1:k, 1:k]^{-1}$. Can we write $Q$ in terms of $P$?

Efficiency aside, you may simply make use of Schur complement. Let $P=\pmatrix{A&B\\ C&D}$, where $A$ is $k\times k$. Then $S=P^{-1}=\pmatrix{(A-BD^{-1}C)^{-1}&\ast\\ \ast&\ast}$. ...
user1551's user avatar
  • 141k
1 vote

Is $\text{Id}-T$ always invertible when $\lim_{n\to\infty} T^n = 0$?

I assume that the space is complete. If $\|T^n\|\to 0$ then $\|T^k\|<1$ for some $k>0.$ Then the operator $I-T^k$ is invertible, by applying the Neumann series. We have $$I-T^k=(I-T)(I+T+\ldots +...
Ryszard Szwarc's user avatar
1 vote

Prove that the given function is invertible

Let \begin{align} x+y+z^2&=v &&\\ x-y+z&=u&&\\ 2x+y-z&=w \end{align} In this system, $v,u,w$ are expressed in terms of $x,y,z$. Can you express $x,y,z$ in terms of $v,u,w$?
user926356's user avatar
  • 1,442
1 vote
Accepted

If $X\sim\mathcal N(x,\Sigma)$, what is $\operatorname E\left[\left\|\Sigma^{-1}(X-x)\right\|^2\right]$?

The law of $X \sim \mathcal{N}(x, \Sigma)$ can be realized via $$ X = x + \Sigma^{1/2} Z $$ where $Z \sim \mathcal{N}(0, I_d)$ and $\Sigma^{1/2}$ is the unique PSD square root of $\Sigma$. Hence, ...
Sangchul Lee's user avatar
1 vote
Accepted

Is this special BLOCK upper triangular matrix diagonalizable?

Since the two diagonal sub-blocks are diagonalisable (because they are Hermitian), $A$ is diagonalisable if and only if it is similar to $B=\pmatrix{A_{1,1}\\ &A_{2,2}}$. Consequently, by Roth’s ...
user1551's user avatar
  • 141k
1 vote

Is this special BLOCK upper triangular matrix diagonalizable?

If $A_{1,1}$ and $A_{2,2}$ have distinct eigenvalues the matrix is straightforwardly diagonalizable since all eigenvalues of $A$ are distinct. If $A_{1,1}$ and $A_{2,2}$ share eigenvalues, it may ...
zinsinho's user avatar
1 vote

The inverse of a lower triangular matrix is lower triangular

Let $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ be either the field of real numbers or the field of complex numbers. Definition (Lower triangular matrix). A (symmetric) lower triangular matrix is a ...
Steven van Dokkum's user avatar

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