Tagged Questions
2
votes
4answers
81 views
Matrix inverse of $\left(A-I\right)$ given $A^{-1}$
I am wondering if the inverse of $$B = A-I$$ can be written in terms of $A^{-1}$ and/or $A$. I am able to accurately compute $A$ and $A^{-1}$, which are very large matrices. Is it possible to ...
0
votes
1answer
52 views
Finding matrix inverse by Gaussian Elimination With Partial Pivoting
Hello guys I am writing program to compute determinant(this part i already did) and Inverse matrix with GEPP. Here problem arises since i have completely no idea how to inverse Matrix using GEPP, i ...
7
votes
1answer
123 views
Inverse of a block matrix
I have a special case where $X=\left(\begin{array}{cc}
A & B\\
C & 0
\end{array}\right)$
and:
$X$ is non-singular
$A$ is singular
$B$ is full column rank
$C$ is full row rank
How do you ...
0
votes
0answers
22 views
Is it possible to simplify matrix multiplication $xM^{-1}y$ using the Matrix Inversion Lemma?
I want to simplify the matrix multiplication $xM^{-1}y$ using the Matrix Inversion Lemma. My aim is to use this identity to convert this multiplication into inverse of a scalar. Is this possible?
...
4
votes
3answers
90 views
Matrix Inverses
So in class we have been discussing matrix inverses and the quickest way that I know of is to get a matrix A, and put it side by side with the identity matrix, like $[A|I_{n}]$ and apply the ...
2
votes
2answers
59 views
Let $A,B$ be elements of $M_2(\mathbb{R})$. Give an example to show that $A+B$ can be invertible if $A,B$ are both non-invertible
The goal for this problem is to show that even if two matrices $A$ and $B$ are non-invertible, $A+B$ can be invertible. I tried to show this using a proof, but I ended up actually proving that this ...
2
votes
2answers
22 views
Let $A_{\alpha}$ be the $\alpha$-rotation matrix. Prove $A_{\alpha}^T = (A_{\alpha})^{-1}$
Let $A_{\alpha}$ be the alpha-rotation matrix. Prove $A_{\alpha}^T = (A_{\alpha})^{-1}$
In other words, prove $A_{\alpha}$ transpose = $A_{\alpha}$ inverse.
First of all, what is a ...
3
votes
1answer
38 views
If $A$ is an $n \times n$ matrix such that $A^3 = O_{3}$, show that $I - A$ is invertible with inverse $I + A + A^2$
So this question is basically a proof.
If $A$ is an $n \times n$ matrix (so square) which satisfies the condition $A^3 = O_{3}$ ($A^{3}$ gives the $3 \times 3$ zero matrix), then show that $(I - A)$ ...
1
vote
0answers
31 views
Sherman-Morrison inverse formula
I've read a paper in which the authors said that they use the "Sherman-Morrison inverse formula". While I know the Sherman-Morrison formula, I couldn't find anything about the inverse of said formula.
...
2
votes
1answer
37 views
Find the inverse for arbitrary k
I need to find a, b, c, d, e, f, g, h (all of which are not zero)
such that for all k is in Real number, show A is invertible or this can't happen
$$A = \left(\begin{array}{ccc}
...
1
vote
1answer
68 views
Inverse of matrix sum, special case: $(A + x I)$
Is there a simple way to do $(A + x I)^{-1}$ where $A$ is an invertible matrix, $I$ is unit matrix and $x$ is a scalar?
I see a lot of expressions for the general case $(A + B)^{-1}$, but nothing on ...
0
votes
1answer
41 views
What is the error in Newton's Method for Matrix Inversion?
I need it to invert a matrix. Wikipedia explains that there is a generalization of the Newton Method for matrices. However, there is nothing mentioned about the error bounds.
Suppose we have, as ...
1
vote
1answer
77 views
Linear Algebra: Least-Squares Approximation & “Normal Equation”
I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] ...
3
votes
4answers
77 views
Is there a good intuitive way to understand why matrix B is inverse of A when matrix A|I is turned into I|B
I'm looking for some help with my intuition of basic matrix operations, specifically finding a matrix's inverse (as per my subject line). I have no problems with the steps. The basic row operations ...
2
votes
1answer
124 views
Power series of matrix which is multiplied by a constant factor $c<1$?
(Important: THIS PROBLEM IS NOT DUPLICATED! Note that the case where just one row of $W$ is multiplied by constant $c$, can be handled by the Sherman-Morrison theorem, but the case where the whole ...
1
vote
0answers
58 views
Easy but hard question about Matrix power series! [duplicate]
Assume $W$ is $n\times n$ matrix and $r<1$ is a real number. Let $$Q = \sum_{i=0}^{\infty} (rW)^i=[I_n-rW]^{-1}$$
Now assume that the matrix $W$ is multiplied by a constant real number $c<1$. ...
0
votes
2answers
55 views
Transpose of matrix inverse: $(AA^T)^{-1}A^Tb \stackrel{?}{=} (A^TA)^{-1}A^Tb$
Given the matrix equation:
$$ x^TA^TA = b^TA $$
I'm trying to find the least squares solution (i.e.; trying to minimize $r=||Ax-b||$). The matrix $A$ is not necessarily symmetric.
When I solve it ...
8
votes
6answers
625 views
If $A^2$ is invertible, then $A$ is also invertible?
True or False: If $A^2$ is invertible, then $A$ is also invertible.
($A$ is a matrix here.)
The answer is true. I was trying to come up with an example that makes this false.
But I couldn't. ...
1
vote
1answer
73 views
Question related to diagonally dominant matrix
A matrix is said to be positive if each entry in the matrix is positive.
If $A$ is real, irreducible, diagonally dominant (or strictly dominant matrix) and has positive diagonal and non-positive ...
0
votes
1answer
43 views
Finding upper triangular matrix
I have this question, and im not sure I know how to solve it.
"Find an upper triangular $U$ (not diagonal) with $U^2 = I$ which gives $U=U^{-1}$".
Anybody who can help me getting the first steps of ...
2
votes
1answer
99 views
If $A$ is an invertible skew-symmetric matrix, then prove $A^{-1}$ is also skew symmetric
Let $A$ be an invertible skew-symmetric $(2n \times 2n)$-matrix. Prove that $A^{-1}$ is also skew-symmetric. (You may assume that $(AB)^T = B^TA^T$).
I did this with a $2 \times 2$ matrix and got ...
6
votes
1answer
80 views
My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible
Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial.
Here's my ...
0
votes
0answers
58 views
Relation between the block inverse and the inverseof the matrix itself?
I have been trying to solve the relation between the block inverse and the inverse of the matrix itself. Hopefully I can get some insights here.
Consider the following vector x consists of the two ...
2
votes
0answers
44 views
Proving invertibility of matrices using banachs lemma
I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...
4
votes
1answer
76 views
Inverse of matrices with 3 parts!
I just wonder if there is any closed form solution for the inverse of matrices with following form, or if it's possible to decompose them.
$
\left[\begin{array}{cccccccccc}
{\color{red}1} & ...
1
vote
1answer
262 views
Inverse of upper triangular matrix
I have an upper triangular matrix that I want to solve the inverse for.
I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity ...
0
votes
1answer
40 views
Benefit of Drazin inverse
What benefits gives Drazin Inverse?
Physically what it corresponds to?
Thanks much
0
votes
1answer
63 views
relation between size of matrix and condition number
I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
0
votes
1answer
34 views
Square root entries of matrices
How would you simplify something like this?
$((\xi'\omega \xi)^{-1})^{0.5}$
where $\xi$ is a $k \times 1$ matrix, $\omega$ is a $k\times k$ square matrix.
Thank you very much!
Edit: Yes, though ...
1
vote
1answer
50 views
About non-negative matrix
If $W$ is diagonal matrix with each entry $W_{i,i}>0$, $K$ is a
symmetric and positive semi-definite matrix and $K_{i,j}>0$ (actually $K$ is a kernel matrix and calculated from a RBF kernel ...
2
votes
1answer
76 views
transpose of inverse vs inverse of transpose
I can't seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix?
Thanks!
2
votes
0answers
144 views
Optimization problem about large matrices
I'd like to solve the following optimization problem:
Find non-negative scalar $a$, $b$, $c$ to minimize
$\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$
where ...
2
votes
0answers
91 views
Low-rank approximation to the Graph Laplacian matrix of a regular grid.
As mentioned in the title, does anybody know any methods of efficient low-rank approximation $LL^T$ to the Graph Laplacian matrix $A$ corresponding to a square lattice? (except PCA)
0
votes
1answer
97 views
Determine if the matrix is idempotent?
I am dealing with an example to show that the matrix($M = I − X(X'X)^{−1}X'$) is idempotent. X is a matrix with T rows and k columns and I the unit matrix of dimension T. And then to determine the ...
0
votes
2answers
102 views
Matrix Inverse Question
Let $C$ be an invertible 2x2 matrix such that:
$$C^{-1} \cdot \begin{bmatrix}1 \\ 2\end{bmatrix} = \begin{bmatrix}3 \\ 4\end{bmatrix}$$
$$C^{-2} \cdot \begin{bmatrix}9 \\ 5\end{bmatrix} = ...
0
votes
2answers
62 views
Application of Matrix Diagonalization
I'm reading a book about inverse analysis and trying to figure out how the authors do the inversion.
Assume that matrix $C$ is
$$
C
~=~
\begin{bmatrix}
88.53 & -33.60 & -5.33 \\
...
3
votes
1answer
63 views
How to compute $\text{trace}((A+D)^{-1}A)$
Give a diagonal perturbation matrix $D$ (which is not an identity matrix), is there a simple way to compute
$$\text{trace}((A+D)^{-1}A)$$
Or is there a good approximation?
3
votes
2answers
63 views
Given the product of a unitary matrix and an orthogonal matrix, can it be easily inverted _without_ knowing these factors?
Given the product $M$ of a unitary matrix $U$ (i.e. $U^\dagger U=1$) and an orthogonal matrix $O$ (i.e. $O^TO=1$), can it be easily inverted without knowing $U$ and $O$?
Sure enough, if $M=UO$, then ...
1
vote
2answers
38 views
Inverting all values in matrix
Lets say I have a matrix:
$$\left[\begin{array}{cc}
2 & 4 \\
3 & 7 \\
\end{array}\right]
$$
And my maximum range value is $10$, how would I go about creating another matrix that ...
0
votes
2answers
45 views
Reverse rows in a matrix
To rotate a matrix 180 degrees around the center point, what I am planning to do is first transverse the matrix, then reverse the rows and then do it again to produce the final result.
This works and ...
0
votes
1answer
76 views
computing the inverse of a special sparse matrix
Given a high-dimensional symmetric postive-definite matrix with only the main diagonal and several other diagonal (say, 1st, 5th and 100th) above and below the main diagonal to be non-zero and all ...
2
votes
2answers
81 views
How to invert sum of matrices?
Given are two matrices:
$\bf A, \bf B$
We know that matrices $\bf A \neq \bf B$ are invertable, symmetric, positive-definite and of full rank. Is it possible to give the formula for following sum ...
0
votes
0answers
52 views
one problem of Laplacian matrix application
Is there a fast method to compute the diagonals or some specific non-diagonal elements (rather than the entire inverse matrix) of the following inverse matrix
$(\alpha L+R)^{-1}$
where $L$ is a ...
2
votes
1answer
96 views
Inverse of orthogonal projection
I have an $N \times N$ orthogonal projection matrix $P = A^H(AA^H)^{-1}A$ that I'm trying to find the inverse for. I'm using matlab, however, I keep getting the warning "the matrix is close to ...
0
votes
1answer
45 views
Transpose of 2 matrices together
So if I have an $m\times n$ matrix $A$ and I represent that matrix as $\displaystyle A = QR$, how do I write $A^{T}$ (transpose) in terms of the original $\displaystyle QR$? Does it become ...
1
vote
1answer
186 views
Derivative of matrix inverse
I am trying to find the derivative of a matrix with respect to the inverse of the same matrix. The matrix in question is a non singular symmetric matrix. Any thoughts?
3
votes
1answer
236 views
Matrix Pseudo-Inverse using LU Decomposition?
What is the step by step numerical approach to calculate the pseudo-inverse of a matrix with M rows and N columns, using LU decomposition?
So far, I have found this, but it uses singular value ...
1
vote
1answer
382 views
Prove that if A is an invertible matrix, then A*A is Hermitian and positive definite.
If I'm not mistaken, if a matrix M has its conjugate M*=M, then M is Hermitian.
In this case then, am I asked to show that (A*A)*=A*A ? What does it have to do with A being invertible though?
And ...
1
vote
2answers
814 views
Matrix is singular to working precision
I have a problem while evaluating inverse using inv in MATLAB.
My matrix looks like this:
...
3
votes
4answers
128 views
How to show $AB^{-1}A=A$
Let $$A^{n \times n}=\begin{pmatrix} a & b &b & \dots & b \\ b & a &b & \dots & b \\ b & b & a & \dots & b \\ \vdots & \vdots & \vdots & ...
