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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1
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1answer
36 views

How to find a eigenvector with a repeated eigenvalue?

The eigenvalues of my matrix are $x_1= 1$ and $x_2=3$ I get an eigenvector $V = t~[ 4~~~~~~ 3 ~~~~~1 ]^T $ but how can I diagonalize the matrix if I have the same column repeated twice. Should I ...
2
votes
1answer
58 views

Largest eigenvalues of AA' and A'A [on hold]

Prove that for every real matrix $A$, the largest eigenvalue of $A'A$ equals the largest eigenvalue of $AA'$ (where ' means transpose). Thanks!
0
votes
3answers
63 views

If a matrix A square is 0, does it follow that A = 0? [duplicate]

Let A be a square matrix. If $A^2 = 0$, then it follows that $A = 0$. Is there a counterexample for this? If there isn't, what kinds of explanation can I make to justify this statement?
1
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0answers
17 views

Transitivity of a Boolean Matrix?

I'm wondering if there's an easy way of visually telling if a boolean matrix has transitivity? The question in particular is: ...
0
votes
0answers
3 views

Link between the cofactors of two related symmetric positive-definite matrices

Let $S = \left( s_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric positive-definite matrix. Let $\Sigma = \left( \sigma_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric ...
-3
votes
1answer
23 views

Prove whether the statement is true or sometimes false. [on hold]

Prove whether the statement is true or sometimes false. If matrix A has row of zeros, does adj(A) have it also?
0
votes
0answers
11 views

there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range?

Let $X,Y \in {M_{n \times m}}$ have orthonormal column. Also there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range(column space) ?
1
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0answers
31 views
+50

Triangulation of matrices

Suppose that $A$ is some triangularizable matrix in $M_n(\mathbb R)$. The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their ...
-1
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0answers
15 views

inequalities for $tr(AB)$ , where A and B symmetry, positively definite matrix

Let $A$ and $B$ be two symmetry, positively definite $n\times n$ matrix with positive eigenvalue $a_1,...,a_n$ and $b_1,...,b_n$ respectively. What's the relationship between them and $tr(AB)$? Are ...
0
votes
1answer
12 views

$X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?

Let $X,Y \in {M_{n*m}}$ have orthonormal column. Also $X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?
0
votes
0answers
26 views

Solution of a general linear system of equations: 4-term n-equations

I have the following system of equations.... $$y_1 = c_{11} \cdot x_{11} + c_{12} \cdot x_{12} + c_{13} \cdot x_{13} + c_{14} \cdot x_{14}$$ $$y_2 = c_{21} \cdot x_{21} + c_{22} \cdot x_{22} + ...
0
votes
2answers
40 views

Dot and Cross Product Proof: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$

How do you prove that: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$ ? The textbook says as a hint to "first do it for $u=i,j$ and $k$; then write $u-xi+yj+zk$ but I am not sure what that ...
3
votes
2answers
35 views

Unitary Matrices and the Hermitian Adjoint

I saw in a definition for unitary matrices, that for a complex matrix being unitary if $M: \mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ is unitary, or: $\langle Mv, Mw \rangle = \langle v,w \rangle$ ...
2
votes
0answers
53 views

Restoring Bidiagonality to a Matrix in SVD Algorithms

Good Afternoon, I am implementing the Golub-Reinsch SVD algorithm and am having difficulty with a boundary case Given a bidiagonal matrix of the form: $$ \begin{bmatrix} b11 & ...
-2
votes
1answer
26 views

What would be the basic solution of this maximization problem? [on hold]

Maximize $P=40x_1+50x_2$ Subject to $x_1+6x_2 \leq 72$ $x_1+3x_2 \leq45$ $x_1, x_2 \geq0$
2
votes
2answers
52 views

What's the easiest way to find all $\alpha\in\mathbb{R}$ such that $\tiny\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$ is positive definite?

For which $\alpha\in\mathbb{R}$ is $$C:=\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$$ positive definite, positive semidefinite or indefinite? It seems to be a simple task, but for ...
6
votes
4answers
86 views

For matrices, if $AB=BA$, then does it follow that $B^{2}A=AB^{2}$?

Suppose $AB=BA$ ($A, B$ are $n\times n$ matrices). Does that mean $B^{2}A=AB^{2}$ ? I looked for counter cases and couldn't find any. I tried to prove this by multiplying both sides and comparing, but ...
1
vote
1answer
13 views

diagonalizing a matrix with random elements

Consider the matrix $A = \begin{pmatrix} cY & 0 \\ 2 & 1\end{pmatrix}$, where $c \in \mathbb{R}$ and $Y$ is a random variable that is uniformly distributed over $[0,1]$ (That is, $Y \sim ...
2
votes
1answer
39 views

Question about matrices?

I have been learning about matrices in my math class and I am confused as to how exactly they work. Take this example: $\left(\begin{array}{c c c c c | c} 1 & 4 & 1 & 0 & 0 & ...
1
vote
1answer
46 views

Finding Matrix of Linear Transformation from $R^2 \rightarrow R^2$

Let $T: R^2 \rightarrow R^2$ be given by: $$T(x_1,x_2) = (4x_1 -2x_2, 2x_1 +x_2)$$ And let $$B = \{(1,1), (-1,0)\}$$ be a basis for $R^2$. First, I write down the matrix of $$T = ...
0
votes
0answers
13 views

relation of dim kers of AB and B operators

I try to prove For any matrixes $A_{ms},B_{sn}$ $$\operatorname{rank}{A}+\operatorname{rank}{B}-s\leq\operatorname{rank}{AB}$$ First, as for any $X$ that $BX=0$ also $ABX=0$, that ...
2
votes
2answers
69 views

How do you find the determinant of this $(n-1)\times (n-1)$ matrix?

It's for a proof of Cayley's Formula, I know I'm being dumb and can't see it, how do I find the determinant of this $(n-1)\times (n-1)$ matrix where the diagonal entries are $n-1$ and the off diagonal ...
1
vote
1answer
19 views

A question about unitary block matrix

For $n,m \in \mathbb N$, let $M_{n,m}(\mathbb C)$ denote the set of complex $n \times m$ matrices and put $M_{n}(\mathbb C):=M_{n,n}(\mathbb C)$. For matrices $A \in M_{n}(\mathbb C), B \in ...
1
vote
0answers
13 views

Discrete fractional fourier transform matrix

I am trying to write a matlab code for some calculations based on Discrete fractional fourier transform. in this article: Optimal filtering in fractional Fourier. after equation (7) a notation Fa is ...
1
vote
3answers
26 views

Trying to figure out formula for deciding how to write Linear Transformation as a matrix relative to a basis

In these lecture notes: http://www.math.rice.edu/~hassett/teaching/221fall05/linalg5.pdf the formula (last line on first page) for finding a matrix relative to bases $B'$ and $B$ is: (1) $$ C_{B'}T ...
-1
votes
1answer
33 views

Find the non trivial solution to a matrix containing a complex number.

$$ A = [B\mid b] = \left[ \begin{array}{cc|c} -3+i & -5 & 0 \\ 2 & 3+i & 0 \end{array} \right] $$ Find the non trivial solution to A. (Solving $B x = b$) I fully understand the ...
2
votes
0answers
36 views

Value of determinant using given conditions.

Let $A$ be a $2$ x $2$ matrix with real entries and $det(A)$ is equal to $d$ which is non-zero. It is given that $det(A +d(adjA))=0$ where $adj$ stands for the adjoint of the matrix. We have to find ...
0
votes
1answer
28 views

How to prove that the column sum for a markov matrix is 1?

As is the topic, it is obvious and easy to explain in non-math language but how do I mathematically prove it?
1
vote
2answers
65 views

Solution to $AB=BA$ given $B$ a $2\times 2$ matrix and non singular?

If $AB=BA$ given $B$ is non singular and a two dimensional matrix can we conclude that $A$ is either a scalar multiple of $B^{-1}$ or a scalar multiple of identity $I$ or a linear combination of ...
5
votes
1answer
92 views

To be a scalar matrix or not to be.

That follows is a pretty (but not so easy) exercise. Is fun off-topic ? Let $A,B\in M_2(\mathbb{C})$. Show that , for every $m,n\in\mathbb{N}$, $((AB)^m-(BA)^m)((AB)^n-(BA)^n)$ is a scalar matrix ...
0
votes
0answers
34 views

Invertible matrix that sets some rows or columns of product to 0 or 1

Assume that there is a fixed binary matrix denoted by $B$ that is a $n\times n$ matrix. " I need to know whether exists a binary matrix $A$ that is both invertible and make matrix $C$ to have some ...
0
votes
0answers
18 views

Generate a function that shuffles a number withing a given range which is reproducible

Lets say I have an array of numbers $1 2 3 4 5 6 7$. I want to shuffle these numbers in some order , $7 5 4 3 1 26$ . However , it should be revesrible. That is given the second array I must be able ...
-2
votes
1answer
21 views

Create matrix of coefficients in matlab

I have written a function that creates a matrix based on a polynomial of a given degree. I will ultimately use this to fit data, but the problem I am having is that I can not find a good way to create ...
1
vote
2answers
55 views

How to calculate the determinants like these?

I'm trying to solve this determinant question and I just can't understand how to approach this. If $x^3$=1, then $$\Delta=\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b ...
0
votes
1answer
39 views

Why is the dimension of the nullspace of A equal to the nullity of A?

For a $m\times n$ matrix $A$, it seems that wikipedia DEFINES nullity(A)=dim(nullspace(A)), whereas my textbook covers the topic the way round: nullity(A) is defined to be n-rank(A). Can one prove ...
0
votes
2answers
41 views

How to minimize $w^{T}Aw$?

$A$ is $n\times n$ matrix. Find a $w$ ( $n$-dimensional unit vector) which minimizes this function. By $w^{T}$, I mean $w$-transpose. I understand there would be non-linear optimization techniques ...
0
votes
0answers
16 views

Problems with the inverse of a banded matrix: not invertible?

I am creating with a software a banded matrix, which is also symmetric. In fact, its definition comes from an array, Array[q], whose length is ...
1
vote
2answers
33 views

Question about matrix multiplication notation

I have the following matrices: $A=\begin{pmatrix} -\frac{2}{3} & \frac{1}{3} & 0 \\ \frac{1}{6} & -\frac{1}{3} & \frac{1}{2} \\ \frac{1}{6} & \frac{1}{3} ...
0
votes
0answers
18 views

Setting up a matrix using logical constraints?

Hello all at Stack Exchange! This is my first post! It took me a while to learn MathJax, but a buddy who referred me said people heavily prefer this format, so I thought I'd just follow the rules of ...
0
votes
0answers
33 views

Algebraic multiplicity of an eigenvalue $λ$

I was going through a question posed on the expression for algebraic multiplicity of an eigen value $\lambda$ on this page : Proving that the algebraic multiplicity of an eigenvalue $\lambda$ is ...
0
votes
0answers
15 views

How to get the projection matrix from coordinate/transformation?

I would like to compare my results with the groundtruth provided by a dataset. For each frame (image) in the groundthruth, I have a projection matrix. For example (for the 0th frame): ...
0
votes
1answer
34 views

What are the names of these variations on the transpose of a matrix and symmetric matrices?

Is there a name for the operator that reflects a matrix over the diagonal running from the top-right to the bottom-left? For the moment, define this reflection of a matrix $A$ as $A^*$. Is there a ...
1
vote
0answers
32 views

Confusion about finding matrix of Linear Transform w.r.t to different bases

I have come across two questions about matrices and changes of bases. They seem to be the same question, but require different approaches. I can't figure out why. First question can be found at: ...
-1
votes
0answers
19 views

Shortest distance and Cross Product [closed]

Show that the shortest distance from a point P to the line through Po with direction vector d is $$ ||P_oP \times d||/||d||$$. I need help writing the proof for this. So far I have: let $ ...
0
votes
0answers
6 views

On the interval minor extremal function of a j × k matrice.

I was going through papers by Marcus/Tardos and Fox and I have this small doubt. If L is a j×k matrix which has every entry equal to 1, what is the interval minor extremal function of L? Can someone ...
1
vote
2answers
25 views

Row Reduce Echelon Form on 3x4 Matrix

I understand the rules for RREF are: 1) Each leading entry must be a 1 in each row 2) Each leading entry's column must be 0's other than the leading entry 3) In stair case order, the next element of ...
0
votes
1answer
42 views

What is known about optimization of spectral properties of matrices over finite fields?

[I am solving the characteristic polynomial over complex numbers but since the matrices are symmetric all eigenvalues are real] Like for symmetric $d-$regular matrices over 0/1 or 0/1/-1 what are ...
0
votes
0answers
13 views

What is or how do you get the rotational matrix of 4-D vector onto the xyz-space?

which would make the 4-D component 0. To be honest I'm not really sure how 4-D rotations work. I know about the simple rotations but not the mechanism in how it rotates, and I'm not sure whether to ...
0
votes
1answer
18 views

Explain $(\|Mx\|_2)^2 = (M^Tx)^T(M^Tx) $ (positive definite, positive semi definite)

Would really appreciate if someone can explain: $$ (\|Mx\|_2)^2 = (M^Tx)^T(M^Tx) $$ can't get my head round with this.
1
vote
0answers
20 views

Find rank of parametered matrix

Today, I've got task: Find all values of parameter a, when rank of matrix M equals 2, where matrix M is 3x3 and has some dependencies on a, for example: $$\begin{pmatrix} 1 & a & a^2-1\\ 1-a ...