Tagged Questions
2
votes
1answer
35 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
$$A = \left(\begin{array}{ccc}
...
1
vote
1answer
62 views
Inverse of matrix sum, special case: $(A + x I)$
Is there a simple way to do $(A + x I)^{-1}$ where $A$ is an invertible matrix, $I$ is unit matrix and $x$ is a scalar?
I see a lot of expressions for the general case $(A + B)^{-1}$, but nothing on ...
0
votes
0answers
25 views
What is the error in Newton's Method for Matrix Inversion?
I need it to invert a matrix. Wikipedia explains that there is a generalization of the Newton Method for matrices. However, there is nothing mentioned about the error bounds.
Suppose we have, as ...
1
vote
1answer
56 views
Linear Algebra: Least-Squares Approximation & “Normal Equation”
I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] ...
3
votes
4answers
71 views
Is there a good intuitive way to understand why matrix B is inverse of A when matrix A|I is turned into I|B
I'm looking for some help with my intuition of basic matrix operations, specifically finding a matrix's inverse (as per my subject line). I have no problems with the steps. The basic row operations ...
2
votes
1answer
118 views
Power series of matrix which is multiplied by a constant factor $c<1$?
(Important: THIS PROBLEM IS NOT DUPLICATED! Note that the case where just one row of $W$ is multiplied by constant $c$, can be handled by the Sherman-Morrison theorem, but the case where the whole ...
1
vote
0answers
57 views
Easy but hard question about Matrix power series! [duplicate]
Assume $W$ is $n\times n$ matrix and $r<1$ is a real number. Let $$Q = \sum_{i=0}^{\infty} (rW)^i=[I_n-rW]^{-1}$$
Now assume that the matrix $W$ is multiplied by a constant real number $c<1$. ...
0
votes
2answers
46 views
Transpose of matrix inverse: $(AA^T)^{-1}A^Tb \stackrel{?}{=} (A^TA)^{-1}A^Tb$
Given the matrix equation:
$$ 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 ...
7
votes
6answers
572 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
65 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
39 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
68 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 ...
5
votes
0answers
56 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
55 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
40 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
74 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
166 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
38 views
Benefit of Drazin inverse
What benefits gives Drazin Inverse?
Physically what it corresponds to?
Thanks much
0
votes
1answer
57 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
58 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
80 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
82 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
73 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
59 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
43 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
71 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
82 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
44 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
158 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
219 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
339 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
651 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 & ...
5
votes
3answers
286 views
If $A$ and$ I+AB$ are invertible, show $I+BA$ is also invertible
Show that if $A$ and $I+AB$ are invertible, then $I+BA$ is also invertible with
$$(I+BA)^{-1} = A^{-1}(I+AB)^{-1}A$$
2
votes
0answers
165 views
How to calculate the submatrix inverse with prior knowledge of matrix inverse?
Given $A\in \mathbb{N}^{n\times n}$, then $A(\mathcal{I})$ is defined by first deleting the those columns with index in $\mathcal{I}$ and then extracting the first $n-|\mathcal{I}|$ rows.
Note that ...
0
votes
2answers
416 views
To invert a Matrix, Condition number should be less than what?
I see that there is a matlab tag in this site, so I ask my question here and not in stackoverflow although it is also related to programming in matlab.
I am going to invert a positive definite matrix ...
2
votes
1answer
72 views
For square matrices $A$, $B$, is $AB=I$ sufficient that $A$ and $B$ are inverse of each other? [duplicate]
Possible Duplicate:
If $AB = I$ then $BA = I$
If $A$ and $B$ are two square matrices, and we know $AB=I$ where $I$ is the identity matrix. Is it sufficient that $BA=I$ as well so that $A$ ...
0
votes
1answer
32 views
Error bound for pseudoinverse
Hi I have a generic matrix A, is it possible to bound the error defined as $\|A^+A−I\|$ ??
Are there some reasonable assumptions (es. random matrix, etc...) I can make in order to have a better bound ...
0
votes
1answer
16 views
Should I check Multicollinearity When There is An Inverse?
At Machine Learning algorithms there are usually inversion process about matrices and sometimes Matlab throws error when Multicollinearity occurs.
Should I check Multicollinearity(and how) everytime ...
0
votes
1answer
151 views
Method of finding inverse of a Matrix using minimal polynomials
Using a piece from my last question I want to show how to find $A^{-1}$ as a polynomial expression in $A$ of degree < $\deg m_A$ where the leading coefficient of the polynomial is ...
1
vote
3answers
530 views
Proving that the matrix is not invertible.
A is a 2x3 matrix and B is a 3x2. How can i prove that the matrix D = AB is not invertible. I could not go further in this problem. The only thing that i have found is the multiply of these two matrix ...
1
vote
1answer
52 views
Invertibility of matrix with each element equal to cofactor
I am doing an exercise book which has one problem that asks you to prove the nonsingularity of a matrix if each element of the matrix equals its cofactor (the determinant submatrix by deleting the ...
