2
votes
3answers
48 views

Find the inverse with respect to the binary operation $a ∗ b = a + b + a^2 b^2$

A binary operation on $\mathbb{R}$: $a * b = a + b + a^2 b^2$ The neutral element I found to be $0$. Then I need to find an invertible element having two distinct inverses. I don't know where to ...
1
vote
1answer
148 views

Linear algebra proof regarding matrices

I'd like a hint rather than a full solution. The problem I am considering is the following: $X$ is an $n\times m$ matrix $Y$ is $m\times n$ Show that $(I - XY)^{-1}\cdot X = X\cdot(I - ...
1
vote
2answers
40 views

for $k\neq 0, -1, 1$, find the inverse of the matrix

for $k\neq 0, -1, 1$, find the inverse of the matrix $$\begin{pmatrix} k&0&0\\ 1&k&1\\ -1&1&k \end{pmatrix}$$ how am I supposed to solve this? all I can think of is plugging ...
0
votes
0answers
14 views

Study the associative and commutative properties and neutral and inverse elements of these groups

Group m*n = max(m,n) on Z and N So i showed its associative by m,n,p in Z and (m*n)*p = max(m,n)p =max(m,n,p) And m(n*p) = m*max(n,p) = max(m,n,p) Commutative m*n = max(m,n) and n*m = max(n,m). I ...
2
votes
1answer
42 views

expansion of matrix inverse

I would like to invert a square matrix $L$. One can write it as a sum of two matrices, one containing the diagonal terms ($D$) and the other the off-diagonal ones ($A$). $$L = D+A$$ I would like ...
2
votes
2answers
62 views

Showing that a matrix is invertible and finding its inverse

I'm incredibly rusty at linear algebra, and in preparation for my course I've been doing some review questions. I've been staring at this one for a half hour and still don't know how to approach it: ...
0
votes
1answer
21 views

Case Deletion Diagnostics

I have NO idea how to approach this problem. I don't see any connection between the corollary and the formula we need to prove. Does anyone have any hints? Corrolary: If $\mathbf{A}$ and ...
1
vote
0answers
16 views

Calculating the left pseudoinverse of a Matrix whose columns are Probablity Mass Functions

I have a matrix $A_{m\times n}$, where $A_j$ , a column of $A$ represents a probability mass function, and so the sum over the column is 1. This is true for all the columns of A, i.e. $\forall j \in ...
2
votes
1answer
32 views

Do lines between determinants pass through the inverse?

Let A be a $2 \times 2$ matrix whose inverse also exists. If I was to draw a line from each of the 3 vertices (that are not the origin) of the determinant of A, to the 3 vertices of the determinant of ...
3
votes
1answer
39 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
3
votes
2answers
109 views

Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$ [closed]

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots ...
0
votes
0answers
29 views

Looking for reference for the criterion of inveribility of a difference of two invertible matrices

It is pretty easy to show that $A-B$ is invertible if either $AB^{-1}$ or $BA^{-1}$ have all eigenvalues of absolute value less than $1$. But I am specifically looking for a handy reference of this ...
0
votes
1answer
70 views

Can an Elementary Matrix's Inverse's Determinant = 0?

Can someone explain to me why an elementary matrix's inverse determinant cannot equal 0? Or can it? Is there some theorem to elementary matrix inverses? THANKS FOR YOUR INSIGHT! :)
0
votes
1answer
20 views

Show that a square matrix with linear transformation T(M) = A·M is bijective when A is inversible

Suppose $K$ is a body (a field), $n ≥ 1$ and $A ∈ M_n(K)$ a fixed matrix. Consider the linear transformation $T : M_n(K) → M_n(K)$ defined by $T(M) = A · M$ for $M ∈ M_n(K)$ The mark scheme says ...
1
vote
1answer
38 views

Proving that there is no invertible matrix with zero row sums using determinants

I have the following question which I know I should use the determinant to solve. Here it is: Determine if there exists an invertible $3\times3$ matrix $A$ such that $$\begin{align*} ...
2
votes
1answer
40 views

Existence of solution for matrix equation $ (I - \alpha A) \bar{x}=\bar{b}$

This is my first question in here and I would be really thankful if someone could help me with understanding the matter. I am solving a matrix equation $(I-\alpha A) \bar{x} = \bar{b}$ for a positive ...
1
vote
2answers
30 views

How to show that a null potent linear transformation is invertible

V is a K vector space and $ψ : V → V$ is a null potent linear transformation i.e. $ψ^N = 0$ for a certain $N ∈ N$. Prove that $Id_V − ψ$ est an invertible element in the ring $L(V, V )$. My assistant ...
0
votes
1answer
29 views

Pseudoinverse and orthogonal projection

Given the matrix $A= \begin {pmatrix} 1 & 1 &1 \\ -1 & 1 & 0 \\ 0 & 2 &1 \end{pmatrix}$. (i) Determine the orthogonal projection $p:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ on ...
1
vote
1answer
31 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
53 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
21 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
39 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
11 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
58 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
49 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 ...
6
votes
1answer
77 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
1answer
39 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
56 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
101 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
608 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
44 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 ...
4
votes
2answers
365 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
43 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
44 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
125 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
27 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
19 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
51 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
46 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
29 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
64 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
77 views

inverse of Vandermonde's Matrix without using determinants

I want to show, that the Vandermonde's Matrix ...
1
vote
2answers
71 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
29 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
45 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
47 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
32 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
37 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} & ...