Tagged Questions
3
votes
2answers
27 views
Linear Algebra determinant and rank relation
True or False?
If the determinant of a $4 \times 4$ matrix $A$ is $4$
then its rank must be $4$.
Is it false or true?
My guess is true, because the matrix $A$ is invertible.
But there is ...
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
$$A = \left(\begin{array}{ccc}
...
1
vote
1answer
62 views
Inverse of matrix sum, special case: $(A + x I)$
Is there a simple way to do $(A + x I)^{-1}$ where $A$ is an invertible matrix, $I$ is unit matrix and $x$ is a scalar?
I see a lot of expressions for the general case $(A + B)^{-1}$, but nothing on ...
1
vote
1answer
33 views
Linear Algebra — Block Matrix Inversion
Please excuse my formatting...
$X=\left(\matrix{A & B\\C & D}\right)$ where $A,B,C,D$ are all $n\times n$ matrices. Assuming that all stated inverses exist show that
...
1
vote
1answer
53 views
Linear Algebra: Least-Squares Approximation & “Normal Equation”
I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] ...
3
votes
4answers
70 views
Is there a good intuitive way to understand why matrix B is inverse of A when matrix A|I is turned into I|B
I'm looking for some help with my intuition of basic matrix operations, specifically finding a matrix's inverse (as per my subject line). I have no problems with the steps. The basic row operations ...
2
votes
1answer
114 views
Power series of matrix which is multiplied by a constant factor $c<1$?
(Important: THIS PROBLEM IS NOT DUPLICATED! Note that the case where just one row of $W$ is multiplied by constant $c$, can be handled by the Sherman-Morrison theorem, but the case where the whole ...
1
vote
0answers
57 views
Easy but hard question about Matrix power series! [duplicate]
Assume $W$ is $n\times n$ matrix and $r<1$ is a real number. Let $$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$. ...
7
votes
6answers
502 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
62 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 ...
4
votes
2answers
32 views
norm of inverse less than 1
I just wanna ask if what I am doing here make sense:
Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
2
votes
1answer
65 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
55 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 ...
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} & ...
0
votes
1answer
29 views
Moore Penrose pseudoinverse, how to show that $A^+\mathbf{b}$ is a solution of $A\mathbf{x}=\mathbf{b}$
$A$ is real valued matrix of size mxn, m > n. SVD decomposition of $A$:
$A=U\Sigma V^T$, where $U$, $V$ are orthogonal and $\Sigma$ is diagonal.
Moore Penrose pseudoinverse $A^+ = V\Sigma^+U^T$. How ...
1
vote
1answer
154 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
37 views
Benefit of Drazin inverse
What benefits gives Drazin Inverse?
Physically what it corresponds to?
Thanks much
2
votes
1answer
47 views
Left inverse of a function
Let $f$ be the function $f\colon \mathbb{N}\rightarrow\mathbb{N}$, defined by rule $f(n)=n^2$.
Needed to find two left inverse functions for $f$. I know only one: it's $g(n)=\sqrt{n}$. Does anyone ...
3
votes
1answer
35 views
does invertibility of product imply invertibility of each term of product?
Suppose $\mathcal{H}$ is a Hilbert space and the product $T_1T_2 \in B(\mathcal{H})$ is invertible. Does this imply that both $T_1, T_2$ are invertible ? I am unable to prove this since, unlike the ...
0
votes
0answers
28 views
Explicit inverse for $U+\Lambda$ with $U$ orthonormal and $\Lambda$ diagonal
I am searching for an explicit expression or at least an efficient way to compute the inverse of $U+\Lambda$ where $U$ orthonormal and $\Lambda$ is diagonal. My attempts so far were very unfruitful.
...
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
2answers
68 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} = ...
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 ...
0
votes
1answer
70 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
51 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 ...
0
votes
0answers
63 views
How I obtain Pseudo Inverse using QR decomposition?
I'm writting a implemetation of linear regression using Normal equation, but I need a pseudo inverse
I found that using QR decomposition and Gauss elimination is easy to obtain the
pseudoinverse.
...
2
votes
1answer
78 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
324 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 ...
3
votes
1answer
131 views
Inverse of a diagonal matrix plus a Kronecker product?
Given two matrices $X$ and $Y$, it's easy to take the inverse of their Kronecker product:
$(X\otimes Y)^{-1} = X^{-1}\otimes Y^{-1}$
Now, suppose we have some diagonal matrix $\Lambda$ (or more ...
4
votes
1answer
84 views
solve $ y = (A+B^{-1})x $ for $x$
I wish to solve numerically for $x$,
$$ y = (A+B^{-1})x $$
with $A, B$ positive definite. So,
$$ x = (A+B^{-1})^{-1}y $$
I would like to avoid calculating $B^{-1}$ since that's generally bad.
...
2
votes
0answers
159 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 ...
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 $A$ ...
1
vote
3answers
83 views
Is this an invertible linear transformation?
Suppose you have a linear transformation $T: M_{2\times 2}\to M_{2\times 2}$ given by
$$ \begin{pmatrix} a & b \\ c & d\end{pmatrix}\mapsto \begin{pmatrix} a+b & a \\ c & ...
0
votes
1answer
32 views
Error bound for pseudoinverse
Hi I have a generic matrix A, is it possible to bound the error defined as $\|A^+A−I\|$ ??
Are there some reasonable assumptions (es. random matrix, etc...) I can make in order to have a better bound ...
0
votes
1answer
148 views
Method of finding inverse of a Matrix using minimal polynomials
Using a piece from my last question I want to show how to find $A^{-1}$ as a polynomial expression in $A$ of degree < $\deg m_A$ where the leading coefficient of the polynomial is ...
1
vote
2answers
72 views
inverse of a binomial matrix
I have a matrix $A$ ($n \times n$) defined as follows:
$$A = \{ 0 \text{ if } i<j,\ \mathrm{Binom}(x=i, \mathrm{size}=j, \mathrm{prob})\text{ if } j \ge i\}$$
This is an upper triangular matrix, ...
2
votes
1answer
138 views
Linear Algebra Question ( rank of matrix )
Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively
prove $\operatorname{rank}(\mathbf{PA}) = ...
0
votes
1answer
24 views
intuitive explanation of sparsity / references
I know it is a vague question, but I am confused by why/when we actually want sparsity of a matrix. For example, interior-point methods work better when constraint matrix is sparse. Similarly, it is ...
1
vote
0answers
81 views
How does adding extra row and column of ones affect a matrix's inverse?
I'm working on a homework problem, and I'm stuck. I guess my linear algebra still needs some work...
I've arrived at
$\mathbf{D}=
\left[
\begin{matrix}
\mathbf{C} & \mathbf{1}^T \\
\mathbf{1} ...
4
votes
2answers
273 views
Inverse of symmetric matrix $M = A A^\top$
I have a matrix, generated by the product of a non-square matrix with its own transpose:
$$M = A A^\top.$$
I need the inverse of $M$, assuming $\det(M) \neq 0$.
Given the nature of the matrix $M$, ...
3
votes
1answer
230 views
How to invert a very regular banded Toeplitz matrix?
What's the best way to invert a simple Toeplitz matrix of the following form?
$$
A = \begin{bmatrix} 1 & a & 0 & \ldots & \ldots & 0 \\\
a & 1 & a & \ddots & ...
1
vote
0answers
101 views
Large number of Linear equation solving with diagonally dominant matrix
For a certain problem I am modelling, I have an MCMC sampler at the moment. It draws samples from the ($n-1$)-dimensional simplex (in this case, from a Dirichlet distribution) and evaluates the ...
0
votes
2answers
72 views
Searching for a suitable invertible function
I am searching for a monotonically increasing and invertible function in $2$ variables. I know several monotonically increasing functions. This is also true for invertible functions. But I am ...
2
votes
2answers
129 views
Show matrix $A+5B$ has an inverse with integer entries given the following conditions
Let $A$ and $B$ be 2×2 matrices with integer entries such that
each of $A$, $A + B$, $A + 2B$, $A + 3B$, $A + 4B$ has an inverse with integer
entries. Show that the same is true for $A + 5B$.
1
vote
1answer
181 views
Prove inequality with norms and matrices
Prove that if $A$ is invertible and $||A-B||<||A^{-1}||^{-1}$ then
$$\lVert A^{-1} - B^{-1}\rVert \leq \lVert A^{-1}\rVert \frac{\lVert I-A^{-1}B\rVert}{1-\lVert I-A^{-1}B\rVert}.$$
I also need ...
