For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0
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0answers
26 views

Using the Four Corners of a square how to I then transform that onto any other Square?

I want to derive a function such that when I enter the x y coordinate of a 2D space I output the corrected values of a square that is offset, potentially sheared and scaled. In my main coordinate ...
2
votes
2answers
28 views

LU-factorization: why can't I get a unit lower triangular matrix?

I want to find an $LU$-factorization of the following matrix: \begin{align*} A = \begin{pmatrix} 3 & -6 & 9 \\ -2 & 7 & -2 \\ 0 & 1 & 5 \end{pmatrix} \end{align*} This matrix ...
0
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0answers
16 views

Graph of Network-Flow Matrices

This matrix is total unimodular 1 1 1 -1 0 0 0 1 1 0 -1 0 0 0 1 0 0 -1 I've read that nearly all ...
1
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0answers
18 views

sequence of positive definite matrices

Let $k\geq 1$ a fixed integer and we put $u_n=\frac{1}{n+k}$ for all $n\in\mathbb{N}$ and we consider the sequence of matrices $$ M_0=(u_0) $$ and $$ M_1=\left(\begin{matrix}u_0 & u_1 \\ u_1 ...
5
votes
1answer
38 views

How to prove that this matrix is total unimodular

This matrix is total unimodular (tested by a computer program). ...
-1
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0answers
32 views

How to rotate two 2D ellipses such that they have maximum cross corelation?

Suppose I have two matrices $A=\begin{pmatrix}3 & 1 \\ 1 & 4\end{pmatrix}$ and $B=\begin{pmatrix}5 &-2 \\ -2 & 4\end{pmatrix}$, where $A$ and $B$ represent covariance ellipses in 2D. ...
0
votes
1answer
27 views

Name of the following property for matrix multiplication.

${\bf Property}$: Let $A$ and $B$ two matrices such that $AB=\alpha I$, where alpha is a real number and $I$ is the identity matrix. Somebody know what is the name of this property? Is there some ...
6
votes
3answers
76 views

If $A^n = B$ and I know $B$, can I find $A$? [duplicate]

Suppose that $A$ and $B$ are invertible, $p \times p$ matrices. If $A^n = B$ and I know all of the entries in $B$, can I find an $A$ for some or all integers $n \ge 0$? How many solutions for $A$ ...
50
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7answers
4k views

Why does this “miracle method” for matrix inversion work?

Recently, I answered this question about matrix invertibility using a solution technique I called a "miracle method." The question and answer are reproduced below: Problem: Let $A$ be a matrix ...
0
votes
0answers
30 views

Diagonalizing to solve a linear recurrence with complex eigenvalues

I know how to solve for a closed form of linear recurrences whose matrix form has all real eigenvalues. What is the difference when solving one with complex eigenvalues? I can't seem to get this ...
0
votes
1answer
18 views

How does banach fixed point theorem related to matrix analysis?

As stated by Banach fixed point theorem, a contraction mapping has only one fixed point. In plain words it means that the contraction mapping T has only one solution that satisfy $Tx = x$. A ...
0
votes
2answers
47 views

$ T^2=T_0 \iff R(T) \subseteq N(T) $. How is it possible? Contradiction.

1) There is excersise 11 in chapter 2.3 of the book "Linear algebra" by Friedberg. In the excersise is stated: Let $V$ be a vector space, and let $T: V \rightarrow V$ be linear. Prove that $ T^2 = ...
0
votes
3answers
29 views

$A,B$ are matrices $3x3$ so that $B^2A=-2B^3$ and $B^3+AB^2=3I$ express $A^{-1}$ and $B^{-1}$ using $B$

I have the follow question : $A,B$ are matrices $3x3$ so that $B^2A=-2B^3$ and $B^3+AB^2=3I$ express $A^{-1}$ and $B^{-1}$ using $B$ I tried to "play" with the equations but I always get stuck with ...
1
vote
1answer
20 views

Power iteration method for computing the largest eigen value

I am trying to self learn the Power Iteration algorithm for computing the largest eigen vector and eigen value. I understood that the algorithm works as follows. Assume we are trying to find the ...
3
votes
1answer
49 views

Why we need Invertible Matrices

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or non degenerate) if there exists an n-by-n square matrix B such that $$A B = B A=I_n$$ I know the definition. ...
0
votes
0answers
15 views

Product of matrices with complex dimensions

In a paper that I have read; there was the following matrix multiplication $ Y= \begin{bmatrix} H_1 \\ H_2 \end{bmatrix} \begin{bmatrix} A_1 & 0 \\ 0 &A_2 ...
1
vote
2answers
28 views

Computational Complexity

"Why are additions known to be cheaper than multiplications?" In contexts pertaining to algebraic complexity theory, this statement is often cited. Can someone elaborate on this? I don't understand ...
1
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1answer
29 views

Number of solutions in system of linear equations

I'm studying System of linear equations. When solving Ax=b, it is said that the system can behave in 3 ways. No solution Unique solution Infinitely many ...
0
votes
4answers
48 views

Applications of 'Matrix Multiplication' [closed]

Where all are matrix products actually needed? (Or perhaps, the practical applications of matrix products in physics,engineering,etc.) I found a few. But, could anyone suggest me more? (Kindly cite ...
1
vote
2answers
31 views

Find trace of matrix M^k in optimum way.

I have to find the trace of every matrix $M^1,M^2.....M^k$ in optimum way. One way is to multiply $M$ every time but complexity increases to $k*n^3$. Is there any better approach?
0
votes
1answer
8 views

Matrix transformations 2-D

So I have a question regarding matrix transformations. I am trying to transform a parallelogram into the unit square. I need to find a series of matrices which transforms each point. For example, ...
0
votes
0answers
13 views

General questions regarding SVD decomposition of matrices

The Singular Value decomposition of a matrix is $$A= U\Sigma V^H$$ I was reading a paper and the paper stated that the SVD was $$A'= V\Sigma V^H$$ 1- I am wondering how come the right singular ...
1
vote
1answer
27 views

positive definite matrix as a function of polynomails

Given polynomials functions as ${f_i}\left( x \right) = \sum\limits_{j = 0}^N {{a_j}{x^j}}$, how can I prove that the following matrix is positive definite or not? $${\bf{A}} = \left[ ...
0
votes
2answers
58 views

Gradient of a Frobenium norm cost Function

Folks - Please help. What's the gradient for the cost function below? $ D(Y||AX)=\frac{1}{2} ||Y-AX||^2_F $ Additional info - -need to get the derivative of that with respect to A. -Multiplicative ...
6
votes
3answers
128 views

Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$

I have the following question : Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ I managed to proof that $I+BA$ invertible My proof : We know that $AB$ and $BA$ ...
0
votes
2answers
17 views

matrix elementary column operations

Till now i was using the elementry row operations to do the gaussian elemination or to calculate the inverse of a matrix. As i started learning the Laplace's transformation to calculate the ...
0
votes
2answers
28 views

Question about eigenvectors of real matrix with real eigenvalues

I have two related questions: Can a real matrix with real eigenvalues have complex eigenvectors? Is it always the case that a real matrix with real eigenvalues is diagonalisable?
5
votes
3answers
50 views

Line for set of three-dimensional vectors

If there is a set for 3D vectors $v$ where $ v \times \begin{pmatrix} -1 \\ 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ -27 \\ 8 \end{pmatrix}$ is a line, what is this line's equation? I'm not sure ...
2
votes
1answer
54 views

Find $a$ in the following matrix

I have the following question : matrix $A$ isn't diagonalizable while $a \in R$ $$A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & a & a-2 \\ 0 & -2 & 0 \end{pmatrix}$$ Find $a$. I ...
2
votes
1answer
29 views

Real eigenvalues, similar symmetric matrix

I know that symmetric matrices have real eigenvalues, and that non-symmetric matrices that are similar to symmetric matrices must also have real eigenvalues, but is the converse true? That is, if a ...
-2
votes
0answers
21 views

pseudospectra of matrix polynomial [on hold]

Definitions: ${A_j},{\Delta _j} \in {\Bbb C^{n \times n}} \ (j = 0,1,2, \dots ,m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + \dots + {A_1}\lambda + {A_0}$ is a matrix polynomial, and ...
-1
votes
1answer
43 views

How do I solve this prove of matrix? [closed]

Let $Ax = 0$ be a homogeneous system of $n$ linear equations in $n$ unknowns that has only the trivial solution. Prove that if $k$ is any positive integer, then the system $A^k x = 0$ also has only ...
2
votes
1answer
46 views

Is this subset of GL(C) a sub-group?

$$G\:=\:GL_2\left(R\right)$$ $$\:N=\left\{A\in G\:;\:A\cdot A^T=I\right\}$$ Is N a subgroup of G? So the main work here is to prove closure for the opposite. We want $A^{-1}\in ...
1
vote
2answers
41 views

Solution of System of linear equations

I have three equations: $$ \begin{cases} 4y + z = 2\\ 2x + 6y - 2z = 3\\ 4x + 8y - 5z = 4 \end{cases} $$ Applying Gauss elimination I get: $$ \left[ \begin{array}{ccc|c} ...
1
vote
1answer
31 views

Commutativity in terms of the Jordan Normal Form.

Let us consider requirements for commutativity of matrices in terms of the Jordan Normal Form, Say we have two matrices $\bf A$ and $\bf B$. Then ${\bf A} = {\bf S}^{-1}{\bf JS}$, where $\bf J$ can be ...
0
votes
1answer
13 views

Smallest dimension for having two non similar matrices with same minimal and characteristic polynomials

I give here the example of two non similar matrices, namely $$M=\begin{pmatrix} 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1 ...
4
votes
0answers
43 views

Is it true that any surjective group homomorphism from $SL_n(\mathbb Z)$ to itself is bijective ? [closed]

Is it true that any surjective group homomorphism from $SL_n(\mathbb Z)$ to $SL_n(\mathbb Z)$ is bijective ?
-1
votes
1answer
34 views

Can $S_n$ be embedded in $GL_{n-1}(\mathbb Z)$ ? [closed]

How to prove that $S_n$ is isomorphic with a subgroup of $GL_{n-1}(\mathbb Z)$ ?
2
votes
2answers
27 views

$\mathrm{Z}(\mathfrak{gl}(2,\Bbb F))$ where the Lie bracket is $[X,Y]=XY-YX$

I want to find $\mathrm{Z}(\mathfrak{gl}(2,\Bbb F))$ where the Lie bracket is $[X,Y]=XY-YX$ So then this will depend on the field, but no harm in direct computation for arbitrary matrices: ...
1
vote
3answers
42 views

Given non-invertible square $A$, find a square matrix $B$ so that $BA$ is invertible.

I have one not invertible $n\times n$-matrix $A$ and want to know how can I find suitable square matrix $B$ that product of $BA$ is invertible.
0
votes
1answer
27 views

Understanding matrix conditioning at a glance

I'm working on my Master's, in the field of computational fluid dynamics, where we have to solve the system $Ax = b$ a lot. As far as I can figure out, if the matrix $A$ is ill-conditioned it is ...
0
votes
1answer
67 views

What is mean by basis of a vector space?

My apologies for asking ambiguous question. [Edited]What i know about basis of a vector space/subspace($\mathbb{R}^n$) is the bunch of vectors $v_{1},v_{2},...,v_{1}$ such that 1) They are ...
5
votes
1answer
42 views

Is there any interesting relationship between a Hermitian matrix and its corresponding entrywise absolute?

In general, a Hermitian matrix can have complex off-diagonal terms. Given any Hermitian matrix $[A]_{n,m}$, I can construct another matrix $[\vert A\vert ]_{n,m} =\vert A_{n,m} \vert$. I would like to ...
0
votes
1answer
21 views

Reduced Row Echelon Form with a Variable

For what value of k does the system of equations not have a unique solution? $$ \left\{ \begin{array}{c} x-2y+2z=0 \\ 2x+ky-z=3 \\ x-y+3z=-5 \end{array} \right. $$ I know that this means I have ...
0
votes
0answers
10 views

Intersection of 2 continua with an arbitrary number of dimensions using matrices

If I have 2 continua (i.e. a line, plane, space, hyperspace etc.) of an arbitrary number of dimensions, each described parametrically as: x = xs + x1*t + x2*u + x3*v + ... + xn*k y = ys + y1*t + ...
3
votes
1answer
76 views

Show that there are numbers c and d such that F(A) = cTr(A^2) + d(Tr(A))^2,

Suppose F(A) is a quadratic function of a real symmetric matrix, A. This means that there are numbers $f_{ijkl}$ so that F(A) = $\sum_{ijkl}f_{ijkl}a_{ij}a_{kl}$. Suppose that $F(A) = F(QAQ^t)$ for ...
0
votes
1answer
66 views

Can't orthogonally diagonalise this symmetric matrix.

So I have a symmetric matrix A = $\begin{bmatrix} 2 & 2 & -4 \\ 2 & -1 & -2 \\ -4 & -2 & 2 \end{bmatrix}$ and I want to orthogonally diagonalise it. I know that there are ...
-3
votes
0answers
27 views

Finding the range and kernel of a matrix [on hold]

How do I workout the Range and Kernel of the following matrix: $\begin{bmatrix}1 & 0 & 1 & 1\\0 & 1 & 0 & 1\\1 & 1 &1 &2\end{bmatrix}$ Please help.
9
votes
2answers
79 views

Exists polynomial satisfying following?

Let $s, u \in M_m(\mathbb{k})$ be a pair of commuting matrices such that $s$ is a diagonal matrix and $u$ is a strictly triangular matrix (with zeros on the diagonal). Put $a = s + u$. Does there ...
0
votes
0answers
20 views

How to derive one matrix algebra equation from another

I have one matrix equation: s = A' X^-1 A (where s is scalar, A is a vector, A' is its transpose, and X^-1 is the inverse of a square, symmetric matrix) Which can be transformed into another ...