1
vote
1answer
28 views

Express summation in terms of matrix norm

Express the following $$\sum _{ i=1 }^{ n }{ ({ \beta }_{ 1 }x_{ i }+{ \beta }_{ 0 }-y_{ i })^{ 2 } }$$ To become something of the form: $∥Ax−b∥^{ 2 }$ where $A$ is an $m$−by−$n$ matrix and $b$ is ...
0
votes
1answer
51 views

Fast way to inverse B'CB+D

$\mathbf {A = B'CB}$, where $\mathbf A$ is of dimension $n \times n$, $\mathbf C$ is m by m, positive definite and symmetric, $\mathbf B$ is of dimension $m \times n$, and $n >> m$. Inversion ...
0
votes
1answer
20 views

Find the point of $f^{-1}$ corresponding to the value of x indicated

i am having problems understanding this problem. The given function $f$ is one-to-one. Find $f^{-1}$, find the point on the graph of $f^{-1}$ corresponding to the indicated value of $x$ in the ...
1
vote
1answer
37 views

Finding a matrix projecting vectors onto column space

I can't find $P$, for vectors you can do $P = A(A^{T}A)^{-1}A^T$. But here its not working because matrices have dimensions that can't multiply or divide. help
0
votes
0answers
7 views

Solving Toeplitz plus Diagonal System, how should I make use of the structure?

I learned that a Toeplitz system, $Ax = B$ where $A$ is Teoplitz, can be solved in $O(n \log n)$ time using Superfast method. or approximate $A$ similar to Approximation method. I am keep ...
0
votes
0answers
27 views

Matrix inverses on matlab: are pinv and ./ related?

I faced with two actual implementations of the same problem, and need some help to find which one is correct. Let K be an non-square $m \times n$ matrix (a product of two eigenvalues vectors), B an ...
0
votes
1answer
47 views

Show that a linear mapping is invertible over all $\Bbb R^{2}$

Show that (under appropriate assumptions) a general linear mapping $F(x,y) = (ax+by,cx+dy)$ is invertible over all of $\Bbb R^2$ (i.e. there is a single inverse for all of $\Bbb R^2$). What ...
5
votes
1answer
62 views

Some questions about the pseudoinverse of a matrix

For every mxn-matrix A with real entries, there exist a unique nxm-matrix B, also with real entries, such that $$ABA = A$$ $$BAB = B$$ $$AB = (AB)^T$$ $$BA = (BA)^T$$ B is called the pseudoinverse ...
0
votes
0answers
22 views

Lipschitz continuity of inverse

Given a function f : $\mathbb{R}^n\to\mathbb{R}^m$, which is known to be Lipschitz continuous, can we say anything about the Lipschitz continuity of it's inverse function (in this case, the ...
0
votes
1answer
48 views

Matrix inverse and Change of basis

I have 2 Change of Basis Matrices $ S_{A,B} $ and $ S_{A,C}$ I want determinate $ S_{C,B} $ We know that $$ S_{A,B} S_{B,C} = S_{A,C} $$ $$ S_{B,C} = S_{A,B}^{-1} S_{A,C} $$ Now i'm quite not ...
1
vote
4answers
98 views

If $B$ the inverse matrix of $A^2$ show that the inverse of $A$ is $AB$

How do I continue from $A(AB) = (BA)A = I$ and how can we justify it if we don't know that $AB=BA$? EDIT: Also, how can we prove that if $AB=I$ then $ BA = I$? This is seperate from the question ...
4
votes
4answers
546 views

Is every self-inverse matrix diagonalizable?

If $A=A^{-1}$, is there always a matrix C such that $C^{-1}AC$ is a diagonal matrix (containing only -1 and 1 in the main diagonal) ? How can I check with PARI/GP, if a given matrix is ...
2
votes
0answers
42 views

Self-inverse matrices with integers with pairwise different absolut values.

Let A be a self-inverse matrix ($A=A^{-1}$) with integer values such that no two integers have the same absolut value. Let M be the maximum of the absolut values (maximum-norm) of A. Which M is the ...
3
votes
2answers
350 views

What can be said about a matrix which is both symmetric and orthogonal?

I tried to find matrices A, which are both orthogonal and symmetric, this means $A=A^{-1}=A^T$. I only found very special examples like I, -I or the matrix $$\begin{pmatrix} 0 &0& -1\\ ...
1
vote
2answers
41 views

What is wrong in the following calculation for the inverse of a matrix?

$\left[\begin{array}{ccc|ccc} 0 & 3 & 0 & 1 & 0 & 0\\ 4 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 5 & 0 & 0 & 1 \end{array}\right]$ ...
1
vote
0answers
37 views

Is there an efficient method to find all the self-inverse matrices with integers in a given range?

Given n and a range, for example [-10,10], is there an efficient method to find all nxn-matrices A with integers in the given range, which are self-inverse, that means the equation $A=A^{-1}$ holds ...
5
votes
3answers
122 views

Can a matrix A with the property $A=A^{-1}$ only have the eigenvalues -1 and 1?

If a matrix A has the property $A=A^{-1}$, are the only possible eigenvalues 1 and -1 ? How can the matrices with integer values and the property $A=A^{-1}$ be characterized ? I found out that if ...
0
votes
0answers
22 views

is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
0
votes
1answer
18 views

Inverse matrix - transformation

I am finding inverse matrix $A^{-1}$ and I was given hint that I could firstly find inverse matrix to matrix B which is transformed from A. $$A=\begin{pmatrix}1 &3 & 9& 27\\3 & 3 & ...
0
votes
2answers
32 views

Method for Finding Matrix-Inverse Through Gauss-Jordan?

When trying to find the inverse of the n$\times$n matrix $A$, one way of going about it is by solving $AX=I$, wherein $I$ is the n$\times$n identity matrix, and $X$ is some n$\times$n matrix which is ...
3
votes
0answers
42 views

Moore-Penrose Pseudo-inverse of a matrix on adding 1 new row/column

Given that I know the pseudo-inverse of a matrix(not necessarily a square matrix), how to calculate the pseudo-inverse of the matrix I get by adding a single row/column to the original matrix? i.e, ...
0
votes
0answers
26 views

determinant and trace of a huge positive definite matrix

I have a problem to compute the determinant and the trace of inverse matrix: $det(\Gamma^{-1}+I_n⊗\Phi^T\Phi)$ and $tr[(\Gamma^{-1}+I_n⊗\Phi^T\Phi)^{-1}]$ where $\Gamma$ is a huge positive definite ...
1
vote
1answer
55 views

Upper bound on the inverse of a Grammian matrix

I have been trying to find a reasonable upper bound on the following: Given $n\in N$ and the Grammian matrix $A_n$ = (($f(i)$ , $f(j)$)) , $f(\lambda) = e^{\lambda t}$ for $0\le t \le 1$ and ...
1
vote
0answers
74 views

inverse of Vandermonde's Matrix without using determinants

I want to show, that the Vandermonde's Matrix ...
1
vote
2answers
36 views

Explicit formula for inverse of upper triangular matrix inverse

I have $n \times n$ upper triangular matrix $A$ such as $$ \begin{bmatrix} x_1 & x_2 & \ldots & x_n \\ 0 & x_1 & \ldots & x_{n-1} \\ \vdots & \vdots & ...
1
vote
2answers
27 views

Inverse function (basic algbra math)

Consider the following function: $f(x) = {1 / (x-6) }$ Find a formula for the inverse of the function. Here is what have so far? $y = 1/(x-6)$ ---> $ x = 1/(y-6) $ But my embarrassing problem is ...
3
votes
0answers
40 views

The inverse of a transpose matrix to “cancel” the transpose?

When it comes to solving and equation containing matrices I don't always understand some of the rules involved. In particular, I am trying to figure out the derivation of the Gauss-Newton algorithm. ...
2
votes
1answer
26 views

relationships of symmetric matrices

I came across the following relationships, but I have no idea how to prove them. I would love to know they can be proved. Suppose $X$ and $Y$ are both symmetric matrices, relationship: $$(X + ...
1
vote
1answer
34 views

Is every invertible matrix over an algebraically closed field diagonalisable?

In $\Bbb{R}$ the only invertible matrices (I can think of) that are not diagonalisable are those which stand for a rotation, but in $\Bbb{C}$ this shouldn't be a problem anymore, since rotations can ...
3
votes
1answer
33 views

Inverse of Cartan matrix

The Cartan matrix of the root system $A_n$ looks like, denote it by $A'_n$ $$A'_n= \begin{bmatrix} 2 & -1 & 0 & 0&\ldots & 0 \\[0.3em] -1 & 2 & -1 ...
1
vote
1answer
31 views

Is there a pseudo inverse $X$ such that $ABX=A$?

Question The title pretty much sums it up. I need to find a matrix $X$ such that: $A B X = A$, with $A\in R^{n\times n}$, $\text{rank}(A)=n$, $B\in \mathbb{R}^{n\times m}$ given. The matrix $X$ ...
1
vote
1answer
32 views

Nonsingular block matrix

Let us consider a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ and the block partitioning $$ \mathbf{A} = \begin{pmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & ...
2
votes
3answers
112 views

Why is a matrix $A$ that fulfils $AA^t = I$ invertible?

Given a square matrix $A$ that fulfils $$AA^t = I$$ Justify why must $A$ be invertible. The answer, according to my book, is simply $$AA^t = I$$ $$A^t = A^{-1}$$ I don't ...
1
vote
2answers
188 views

If a matrix is not invertible, is it still possible to find a left and/or right inverse?

I was recently asked to find the right inverse of some matrixes. I found that all three of them were invertible, so it was just a matter of finding their inverses, which would be exactly the same as ...
2
votes
2answers
20 views

About inverse matrixes

I've been reading about invertible matrixes. I have a few questions: One theorem says The rank of an invertible matrix of size $n$ is $n$. So, is it safe to say that all invertible matrixes ...
0
votes
1answer
57 views

How to prove that a matrix inverse is invertible?

First off, I'm trying to prove that $(A^{-1})^{-1} = A$, but in my proof, I assume that $A^{-1}$ is invertible. I'd like to see or do a proof that $A^{-1}$ must be non-singular, but I'm stuck at ...
1
vote
1answer
61 views

the rank of a matrix and its inverse are always equal

I had a true or false quiz in a linear algebra course, one of the statements read the rank of a matrix and its inverse are ...
3
votes
2answers
134 views

Conditions for a matrix to be invertible

Let $n \geq m$ and let $C$ be a $n \times m$ full rank matrix, that is $rank(C) =m$. Considering that $D$ is a diagonal positive semidefinite matrix, under which conditions is the $ m \times m$ matrix ...
1
vote
1answer
62 views

What does it mean when a matrix is to the (-1/2) power?

I'm reading a machine learning paper that uses a form of matrix normalization called symmetric divisive; given a matrix A and a diagonal matrix D derived from A, we define $$N=D^{-1/2}AD^{-1/2}$$ I am ...
0
votes
2answers
56 views

Linear Algebra Review Questions

So I have a test on Monday and my professor posted a couple of non-graded review questions that she said we should look over. Anyhow, I have a couple of questions that I'd like answered if that's ...
0
votes
0answers
28 views

Inverse of a non-singular linear transformation

The question is about showing that if A is a non-singular linear transformation of an n-dimensional linear space to itself, then there must be some polynomial $c_0 + c_1 z + ... + c_k z^k$ such that ...
3
votes
1answer
110 views

How find this matrix the inverse $A^{-1}$

Let $a,b>0$,and the matrix $A_{n\times n}$ and such $$A=\begin{bmatrix} a&b&0&\cdots&0&0\\ b&a&b&\cdots&0&0\\ 0&b&a&\cdots&0&0\\ ...
0
votes
2answers
46 views

Prove $ X = \left(\begin{array} &A & C \\ 0 & B \end{array} \right) $ is invertible iff A and B both are.

Suppose $A$ is a $n \times n$ matrix, $B$ is a $m \times m$ matrix, and $C$ is a $n \times m$ matrix. Prove $ X = \left(\begin{array} &A & C \\ 0 & B \end{array} \right) $ is invertible ...
1
vote
2answers
123 views

Proof of the Inverse of a Scalar times a Matrix

How would I prove that given a square matrix $A$ and non-zero scalar $c$ that $$(cA)^{-1}=c^{-1}A^{-1}$$
0
votes
2answers
124 views

Power series for a matrix inverse

Is there a power series expansion for a matrix inverse of the form $$\left(\frac{1}{m}I+A\right)^{-1} \mbox{ where $m$ is a scalar?}$$ $A$ is not invertible but the expression above is defined. I ...
1
vote
1answer
43 views

Moore-Penrose inverse multiplication

I really need help proving that when $AB=0$ then $B^+A^+=0$ and also the other way: when $B^+A^+=0$ then $AB=0$. Where $B^+$ and $A^+$ are Moore-Penrose Pseudo-inverse of B and A.
0
votes
1answer
42 views

If A = BC and B is invertible, then how does reducing “B to I” also reduce “A to C”?

If $A = B*C$, where $B$ is an inverse, use row-ops to reduces "$B$ to $I$" also shows that it will reduce "$A$ .. $C$". Big-Hint: Represent the row operations by a sequence of elementary matrices.
1
vote
2answers
259 views

Show that A is invertible and that it is Lower Triangular.

Does anybody have a solution to the given word problem below? Let A be a lower triangular n x n matrix with nonzero entries on the diagonal. Show that A is invertible and and that A-inverse is lower ...
0
votes
0answers
43 views

How to calculate the inverse of a known optical distortion function?

Assume I have the following lens distortion function: $$ x' = x(1 + k_1r^2 + k_2r^4) \\ y' = x(1 + k_1r^2 + k_2r^4) $$ where $r^2=x^2 + y^2$. Given the coefficients $k_1$ and $k_2$ I would need to ...
2
votes
0answers
64 views

Is the pseudoinverse of a singular, lower triangular matrix itself lower triangular?

Suppose $L\in\mathbb{R}^{n\times n}$ is a singular, lower triangular matrix. Is its psuedoinverse, $L^\dagger\in\mathbb{n\times n}$, also lower triangular? I have already proved by induction that the ...