Questions tagged [pseudoinverse]

The operator which best approximates a solution to a linear system with a singular (non-invertible) matrix.: e.g., the Moore-Penrose pseudoinverse. Use when a question concerns a matrix that is probably singular.

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Pseudoinverse of a matrix with columns of exponential decays

I want to calculate the pseudoinverse $A^+$ of a matrix $A$ whose columns are exponential decays: $$ \begin{pmatrix} e^{-\alpha_{0}t_{0}} & e^{-\alpha_{1}t_{0}} & e^{-\alpha_{2}t_{0}} ...
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Are all the conditions of the Moore-Penrose inverse definitions necessary?

The Moore-Penrose inverse of a real or complex matrix $M$ is the unique matrix defined by four conditions. Can any of these four conditions be relaxed with no loss of uniqueness? I noticed there was a ...
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Prove that a particular matrix is positive definite

Let $A$ and $V$ be positive semidefinite matrices, and let $U$ be an orthogonal matrix, partitioned columnwise: $$U = \begin{bmatrix} U_1 & U_2 \end{bmatrix}$$ Consider the matrix $$M = A + U_1^...
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Solving a linear matrix equation with both left and right multiplication of unknown

I would like to solve a matrix equation of the form $$ \mathbf{A} \mathbf{X} + \mathbf{X} \mathbf{A}^T = \mathbf{B} $$ where $\mathbf{A}$ and $\mathbf{B}$ are known $n \times n$ matrices, and $\...
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Pseudoinverse of block diagonal matrix

Suppose I have some block diagonal matrix $A$, defined as: $A = \begin{bmatrix} A_1 & 0 & ... & 0 \\ 0 & A_2 & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 &...
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22 views

Pseudo inverse operation of matrix

Let $h_{i} \in \mathbb{C}^{1\times M}$ and $M$ is a postive constant. $H \in \mathbb{C}^{K\times M}$ is a matrix including $K$ vectors, i.e. $H = [h_{1}^{T},...,h_{K}^{T}]^{T}$, $K<M$. Let $W$ ...
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Pseudo inverse of matrix with special form

Consider $A=BC$. A has orthonormal columns. I want to verify that the inverse $D=C^+ B^+ $ satisfies Penrose properties. The following holds: $ CDC = ABB^+A^*AB = ABB^+B \overset{!}{=} AB = C $ ...
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Pseudo-inverses equivalence

I am wondering what is the relation between Pseudo-inverses and the following: $Ax=b$, when $A$ is singular, then $x = \left( A^T A \right)^{-1} \left(A^T b \right)$. Then, what is the difference ...
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Moore Penrose inverse of the end-point map

We consider the end-point map $\mathcal{E}$ of a nonlinear control system: $$ \dot{x}(t) = f(t,x(t), u(t)), \quad x\in \mathbb{R}^n,\ u\in \mathcal{U}\subset L^{2}([0,T], \mathbb{R}^m) $$ starting ...
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44 views

Question about the Moore-Penrose pseudoinverse

Suppose I have a matrix $A$ that is not invertible, but such that $A + \epsilon I$ is invertible for all $\epsilon$. I'm wondering whether we can say something like $$\underset{\epsilon \to 0}{\lim} (...
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I need help proving something about the pseudoinverse?

I need to prove that for every vector $x ∈ \mathbb{R}^m$, $(AA^+)x$ is the orthogonal projection of $x$ onto $\operatorname{Col}(A)$, where $A^+$ is the Moore-Penrose pseudoinverse of $A$. I don't ...
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Pseudo inverse: What is the best way to solve F from $FA^kB$ and $FA^kG$ if $A, B, G$ are known?

Assume that we know $A, B, G$ and also we know $y^{(1)}_k = FA^kG$ and $y^{(2)}_k = FA^kB$. But I don't know $F$. I want to find $F$. I want to take adventages of both $y^{(1)}_k$ and $y^{(2)}_k$, ...
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Where can I found more information about this kind of “pseudo-inverse”?

I have the following matrix: $$ A = \left[ \begin{array}{llll} +1 &-1 &+0 &+0\\ +0 &+1 &-1 &+0\\ +1 &+0 &+0 &-1\\ \end{array}\right] $$ for which I verified that ...
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How can I solve $A\operatorname{diag}(x)=B\;?$

How can I solve $A\operatorname{diag}(x)=B\;?$ I am actually an engineer and recently involved in this field. Thanks.
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How to solve with Tikhonov regularization or some kind of regularization? Ax = b

This is a classical issue! It have its orgins from Observer Kalman Filter Identification I have tried to solve it with pseudo inverse, that is using Singular Value Decomposition(SVD). I have also ...
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Perturbation analysis and sensitivity of eigenvector matrix product with specific perturbation

In my research in applied linear algebra and probability (Wiener filtering) I have come across this rather interesting problem: For a matrix $ U $ we denote by $ U_k $ the matrix formed by taking ...
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Find a common magnitude for two vectors such that they intersect

Trying to solve for a common magnitude of two non-parallel vectors such that they intersect. I am currently solving using a left inverse, but I am not sure if this is correct. Let $ x_1 = x_1^0 + ...
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On the number of connected functional digraphs with the same iterated preimage structure

I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$. For example, $P_j=\left[f^{-j}(...
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Moore-Penrose pseudoinverse: product on left with another matrix

The following problem comes from studying the conditional expectation of a multivariate normal distribution. Let $n\ge2$ be an integer, and let $\Sigma$ be a positive semidefinite, symmetric $n\times ...
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Pseudochange of Basis

I have a question regarding a change of basis and the pseudoinverse. Let $\mathit{B}_{1}$ and $\mathit{B}_{2}$ be basis for $\mathbb{R}^2$ with square $2 \times 2$ $\mathit{A}$ matrix describing $\...
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Show $AA^+$ is symmetric

Can somebody show me how $AA^+$ is symmetric if $A^+$ is the pseudoinverse of $A$? All I can muster is: $(AA^+)^T => (A^+)^TA^T$ I know: $(A^+)^T = (A^T)^+$ but that doesn't really seem like it ...
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Pseudoinverse of $T: L^2(a,b) \to L^2(a,b)$, $f \mapsto g f$, where $g \in \mathcal{C}(a,b) \setminus \{0\}$

Find the pseudoinverse $T^+$ of $$T: L^2(a,b) \to L^2(a,b), \ f \mapsto g f,$$ where $0 \ne g \in \mathcal{C}(a,b)$. First find $\mathcal{N}(T)$ and $\mathcal{R}(T)$ (which denote kernel and ...
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Exercise problem from Matrix Differential Calculus with application in Statistics and Econometrics

a) Show that $X′V^{−1}X(X′V{−1}X)^{+}X′ = X′$ for any positive definite matrix V. b) Hence show that if $C(R′) ⊂C(X′)$, then $R(X′V^{−1}X)^{+}R′(R(X′V^{−1}X)^{+}R′)^{+}R = R$ for any positive ...
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Problem related to mp inverse.

Show that $X′V^{-1}X(X′V^{−1}X)^{+}X′ = X′$ for any positive definite matrix V. My attempt: I am writing $V^{-1} = GG'$, and then substituting it in the LHS of the above equation. LHS = $X'GG'X(X'...
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Pseudo Inverses of covariance matrices that are close to each other

I am trying to understand inverses of covariance matrices that are close to each other. I start with a positive semidefinite matrix $\mathbf{\Sigma^{-1}}$, then I take the pseudo-inverse to get ...
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45 views

The pseudoinverse of a rank one matrix

Let $A$ be a $m\times n$ matrix with real coefficients. It is proved, for example here Derive the Pseudo Inverse (Moore Penrose) of Rank 1 Matrix as a Scalar Multiple of Its Transpose that if $A$ is a ...
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Show that $A^+=V_rS^{-1}U_r^T$

Let $A^+:=V\sum U^T$ and $A:=USV^T$ Show that $$A^+=V_rS^{-1}U_r^T$$ Where the columns of $V_r, U_r$ are the first $r$ singular vectors. My try: $A^+=\sum_{i=1}^r \frac{1}{\sigma_i}v_iu_i^T,\...
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Least Squares removing first $k$ observations Woodbury formula

Given the matrix $X_{n,p}$ from the least squares problem $$ \mathbf{X} \cdot \mathbf{\beta} = z $$ Where the normal equation is: $$ \mathbf{\hat{\beta}} = \left(\mathbf{X}^T \mathbf{X}\right)^{-1} ...
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Pseudoinverses giving weird results

Suppose H and E are two $n\times m$ matrices with $n>m$, now I have an equation: $$\begin{equation}H=E\rho \end{equation}$$ where $\rho$ is $m\times m$ since $n>m$, we have moore-penrose ...
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In $u^{T}A^{-1}= v^{T}B$ with $B$ being a non-square matrix, why are there infinite solutions for $v$?

I have a matrix equation in the form of $$u^{T}A^{-1}=v^{T}B$$ where $u$ and $v$ are vectors, $A$ is a square matrix, and $B$ is a non-square matrix. I'm told that while there is a unique solution ...
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Moore-Penrose pseudoinverse and the Euclidean norm

Section 2.9 The Moore-Penrose Pseudoinverse of the textbook Deep Learning by Goodfellow, Bengio, and Courville, says the following: Matrix inversion is not defined for matrices that are not square. ...
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Pseudoinverse of $\mathbf{A} \in \mathbb{R}^{m \times n}$ multiplied by $\mathbf{A}$

The Moore-Penrose pseudoinverse of a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ is \begin{equation} \mathbf{A}^+ = (\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T. \end{equation} Now, using this we ...
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What is the pseudoinverse of a singular square matrix?

All matrices have a pseudoinverse. However, I cannot prove this for a singular square matrix. The pseudoinverse of a matrix is given as $$A^{+} = (A^{T}A)^{-1}A^{T}$$ $$(A^{T}A)^{-1} = \frac{C^{T}...
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rank of matrix $A$ and result of Moore-Penrose pseudoinverse $A$ times $A$

Let $A$ be a real $4 \times4$ matrix with rank($A$)= 2 and two columns of zeros as follows $$ A = \begin{bmatrix}a_1&0&b_1&0\\a_2&0&b_2&0\\a_3&0&b_3&0\\a_4&0&...
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Moore-Penrose inverse of Moore-Penrose inverse

If Y is the pseudoinverse of matrix X, then X will be the pseudoinverse of Y. This is a trivial consequence once the Moore-Penrose conditions are written: $Y = X^+$ implies $$ \begin{aligned} XYX&...
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Projector on subspace spanned by linearly dependent vectors

Given a vector $y$ in a Hilbert space (possibly infinite-dimensional) and the subspace $\mathcal{T}$ spanned by a set of linearly dependent vectors $(x_1, \dots, x_p)$, I want to write the orthogonal (...
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Calculating pseudo inverse for specific vector products

Let $v^tv = w^tw = 1$; $v,w \in\mathbb{R}^n$, and $I$ be the identity matrix. Calculate the pseudo inverse $A_k^{+}$ for: $A_1 = v$ $A_2 = v^t$ $A_3 = v^tw$ $A_4 = vw^t$ $A_5 = I - 2vv^t$ This is ...
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Is there a term to describe two matrices that are pseudoinverses of each other?

Given the matrices $A$ and $B$ such that $ B = A^+$ and $ A = B^+$, is there a special term in linear algebra that describes this relation or does one just say each matrix is the pseudoinverse of the ...
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Quantifying the “usefulness” of Penrose-pseudoinverse

Given a real MxN matrix $A$ (non-invertible), is there a way to quantify the degree of "usefulness" of the Penrose-pseudo inverse $A^+$? Is the distance $||AA^+ - I|...
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How do these two matrices relate to each other?

Given $$A = \begin{pmatrix} 0&& 1&& 0&& 0 \\ 0&& 0&& 2&& 0 \\ 0&& 0&& 0&& 3\end{pmatrix}$$ and $$B = \begin{pmatrix} 0&&...
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$\text{cond}_2^2(M)= \text{cond}_2(M^T M)$ for non square matrix

Let $M \in \mathbb{R}^{m \times n}$ be a matrix with full colum rank. Proof $$ \text{cond}_2^2(M)= \text{cond}_2(M^T M). $$ What I got so far: Denote the pseudo-inverse (Moore–Penrose pseudoinverse)...
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Uniqueness of solution for a linear system

Suppose that $A\vec{x} = \vec{0}$ is a linear system. Also columns of $A$ are linearly independent. Prove that $\vec{x} = \vec{0}$ is the solution and it's unique. My answer: If $A$ is a square ...
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how can I calculate the inverse of matrix?

Matrix has 16 rows and 1166 columns so I did not inverse directly. I have to calculate the SVD of matrix. I used the Matlab and I try to calculate pseudo inverse of matrix. [S V D]= svd(A) A= SVD' ...
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if for a Matrix A $ A^3 = 0 $ and A has an Inverse is it safe to assume that A = 0?

I'm sorry if this is an obvious question. I searched and haven't found anyone has asked it before (probably because it's so obvious). I went over some basic exercises and found one that asks me to ...
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Equivalence of two invertibility criteria.

I would like to prove that for two right-invertible matrices $A, B \in \mathbb{R}^{m \times n}$, the following two statements are equivalent (a) $(I - A^+ A) + (I - B^+ B) $ is invertible. (b) $\...
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Recursive approach to multiple linear regression models

Let $$ Y_i = X_i \mathbf{\beta}_i + \epsilon_i , \qquad i = 1 , ... , T, $$ be a finite number of multiple regression models indexed by $i$. Here $X_i$ is the design matrix $m \times k_i$, $\mathbf{\...
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Linear algebra matrix inverse and moore-penrose pseudo inverse

I know, if I have the equation: X = A^-1 * B where A,B,X is element R 100x100 I can change formula to: X^-1 = B^-1 * A But is it also the same, if I have the equation: X = A^-1 * B where A,B is ...
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16 views

Right Inverse from Singular Values

Let $A$ be a matrix with SVD $A = U \Sigma V^*$. Suppose that $$ \Sigma = \begin{bmatrix}1&0&0&0\\ 0&2&0&0\\0&0&3&0\\0&0&0&0\end{bmatrix}.$$ The right-...
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149 views

Moore-Penrose Pseudoinverse of a matrix product

$X_{n \times p}$ is a real, thin ($n>p$) rectangular matrix of rank $p$, so $X^T X$ is full rank. The Moore-Penrose pseudoinverse of $X$ is given by $X^+=(X^TX)^{-1}X^T$. Let's now define $W=XA$ ...
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What is the estimate coefficient when design matrix is singular, when using the Moore-Penrose generalized inverse matrix?

The OLS estimators is But if $Y$ is a $50 \times 1$ matrix, and the design matrix $X$ is $50 \times 16$. The design matrix $X^T X$ is singular, then what will the estimate coefficient function be, ...

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