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
vote
4answers
64 views

why does the reduced row echelon form have the same null space as the original matrix?

What is the proof for this and the intuitive explanation for why the reduced row echelon form have the same null space as the original matrix?
2
votes
2answers
52 views

If some power $A^n$ of a matrix $A$ is symmetric, is $A$ necessarily symmetric?

If $A^{n}$ is a symmetric matrix, should I conclude that A is also symmetric?
0
votes
1answer
16 views

Adding a dependent row to a matrix with LI rows

Lets say my matrix is giving me a unique solution.What if I add another row that is some combination of already present rows?I know it would set the determinant to zero and now the solution may not ...
1
vote
1answer
22 views

Different representations of a matrix in reduced row echelon form

EDIT: I decided to ask this question after working on this particular problem. I had the stupidity to think that row-reduced = row reduced echelon form. Brain fart, nothing more to see here... ...
0
votes
0answers
5 views

Minimize real valued scalar function $f(Q)$, where $Q=diag(\vec{q})$ subject to $q_j\geq0$ $\forall j\in\{1,2,…,m\}$ (positive vector)

Let variable vector $\vec{q}$ of size $m\times1$, and its diagonal counterpart $m\times m$ matrix $Q=diag(\vec{q})$, for some $m\in\mathbb{N}$. Define fixed parameter $n\times1$ vectors $\vec{p}, ...
2
votes
1answer
38 views

Can you add a scalar to a matrix?

If I add a scalar to every element of a matrix, e.g. for a $2\times2$ matrix $$ \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix} + b \overset{?}{=} \begin{pmatrix}a_{11}+b & ...
0
votes
2answers
31 views

Final transformation matrix

I have a 3d object, to which I sequentially apply 3 4x4 transformation matrices, $A$, $B$, and $C$. To generalize, each transformation matrix is determined by the multiplication of a rotation matrix ...
1
vote
1answer
21 views

Matrix inequality $A^2 \succeq A$

If $A$ symmetric positive semidefinite matrix is the following inequality true. If $A \succeq I$ then \begin{align} A^2 & \succeq A \end{align} This is an equivalent of $a^2 \ge a$ is $a \ge ...
0
votes
1answer
22 views

3 Points in 3D Space to Develop an Arc or Circle

Background: I'm a Robotics Engineer and I am trying to develop a more flexible, modular, and robust program for our welding robots, which will minimize teaching time for new robots and also minimize ...
0
votes
4answers
71 views

Prove that $g(A)$ is an invertible matrix

Let $A\in M_n(\mathbb{C})$ and let $\lambda\in\mathbb{C}$. Prove that if $\lambda$ is not an eigenvalue of $A$ then $A-\lambda I$ is invertible. Moreover, for $g(x)\in \mathbb{C}[x]$, prove that if ...
15
votes
4answers
1k views

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
-1
votes
1answer
21 views

Let $ \alpha \neq 0 $ isn't $ n \times n $ identity matrix and $ P $ is $ n \times m $ matrix. Let $ P = \alpha P $ . When $ P \neq 0 $? [on hold]

Let $ \alpha \neq 0 $ isn't $ n \times n $ identity matrix and $ P $ is $ n \times m $ matrix. Let $ P = \alpha P $ . When $ P \neq 0 $ ?
0
votes
0answers
12 views

What does $M_{uv}^l$ represent?

Let $M$ be any non negative square matrix. What does $M_{uv}^l$ represent? $M_{uv}^l$: $uv$ entry of $M^l$. (When $A$ is adjacency matrix of a graph, then $A_{uv}^l$ is number of walks of length $l$ ...
0
votes
0answers
14 views

Representing all pairs shortest path in a graph with a matrix

Given a graph $G(n,E)$ where $n$ is the number of nodes and $E$ represents the edges. Is there a way to represent or transform this into a matrix containing all the shortest paths between two pairs ...
-1
votes
0answers
26 views

Matrix polynomial [on hold]

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

Is it possible to determine if a matrix is not diagonalizable via row operations?

Suppose a matrix can be row reduced to the identity matrix, is this enough to say that it is not diagonalizable? If so, what theorem(s) or logic figures this out?
4
votes
4answers
43 views

Equivalent definitions of an orthogonal matrix.

I wish to show that the following definitions of an $n \times n$ real matrix $Q$ are equivalent: $QQ^T=I$, $Qx\cdot Qx=x\cdot x$ for all $x\in \mathbb{R}^n$. I found it easy to show that ...
-3
votes
0answers
13 views

E as an expectation of a quadratic form [on hold]

if E(expectation of quadratic form) is an operator, show that E(AB+C) = AEB + EC. where b and c are variables.
0
votes
0answers
18 views

Show that every Jordan matrix has a cyclic vector

Is my following reasoning correct? Since an $n\times n$ Jordan matrix has rank $n-1$ (because we can only make the last row the zero row), its geometric multiplicity is 1, which means the matrix has ...
1
vote
2answers
32 views

Technical question in Vandermonde determinat proof

I can follow the proof given in (2nd proof, or the induction proof), until the sentence: "From the Expansion Theorem for Determinants‎, we can see that the coefficient of $x_k$ is:". I don't ...
2
votes
1answer
25 views

Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
4
votes
2answers
71 views

Is this matrix invertible?

I have been working on a proof and am stuck with showing that the below matrix is invertible. I am not interested in the explicit inverse, only showing it has a nonzero determinant as the existence of ...
1
vote
2answers
25 views

spectral radius

Does the spectral radius of a matrix defines a norm? I mean does it satisfy the properties of norm, ie. $$||x|| \ge 0$$ $$||x|| = 0 \implies x=0$$ $$||kx|| = |k|\;||x||$$ $$||x+y||\le ||x||+||y||$$
-2
votes
3answers
30 views

Find the matrix product when possible

$\begin{bmatrix}-1& 3\\ 3 & 4 \end{bmatrix} * \begin{bmatrix}0& -2 & 4\\ 1 & -3 & 2\end{bmatrix} $ I realized that there is no third matrix column so does that mean I assume ...
1
vote
2answers
47 views

Showing that the set of $2 \times 2$ real orthogonal matrices has a particular parameterization

Theorem Every orthogonal matrix in $\mathbb{R}^{2, 2}$ is in the form \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} or \begin{bmatrix} \cos\theta ...
0
votes
4answers
34 views

Find the indicated matrix

$$ A=\begin{bmatrix} 2& 3\\ 2 & 4 \end{bmatrix} $$ $$ B=\begin{bmatrix} 0& 4\\ -1 & 6 \end{bmatrix} $$ Find 2A+B Would I go 2*2+2*3+2*2+ *4+ 0+4+-1+6?
0
votes
0answers
31 views

Maximization of sum of convex functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
0
votes
0answers
25 views

Can I do Gaussian Elimination on this? (mod 2)

I have this matrix in GF(2): [0, 0, 1, 0] [1, 1, 0, 0] [0, 0, 0, 1] It's not a square matrix but I tried to do Gaussian elimination on it anyway after adding a ...
2
votes
1answer
56 views

A question about matrix algebras

Let $A,B \in M_n$, $n \geq 2$. If $A$ and $B$ do not share a common eigenvector, why is $\mathcal{A}(A,B) = M_n$? Notation and definitions: $M_n$: the set of $n \times n$ matrices over ...
1
vote
1answer
19 views

What is the characteristic polynomial of power of a matrix

If the c.p. of A is $(\lambda-2)^3(\lambda+2)^2(\lambda+3)$, how can I find the c.p. of $A^2$? Would it be $(\lambda-4)^3(\lambda+4)^2(\lambda+9)$? Thanks!
0
votes
1answer
35 views

Perform the operation or operations when possible.

$\begin{bmatrix}-5& -9\\ 9 & 3 \end{bmatrix} + \begin{bmatrix}8& 5\\ -4 & -1\end{bmatrix} -\begin{bmatrix} 4& -7\\ -9 & -6 \end{bmatrix} $ Also, I was trying to add ...
3
votes
4answers
63 views

Does an $n\times n$ matrix $A$ only have an inverse if $rank(A)=n$? If so, why?

I'm currently learning about the rank and inverses of matrices, and every time I attempt to find the inverse of a matrix with a rank smaller than it's number of rows, I find I am unable. One example ...
6
votes
2answers
38 views

minimum eigenvalue for difference of two matrices

Let $A$ a symmetric positive definite matrix, and $B$ a matrix constructed from $A$ by setting all its off-diagonal elements to zero. Then is there a way to see for which values of positive scalars ...
0
votes
0answers
19 views

Asymptotical stability of a discrete dynamical system

There is a linear time invariant discrete system, \begin{align} x_{k+1}&=\tilde{A}x_k, \end{align} where $\tilde{A}$ is a block matrix represented by \begin{align} \tilde{A}= ...
1
vote
0answers
14 views

Determine mutual location of two coordinate systems, given two sets of points

My problem is: we've got tracking device and a robot. Tracking device provides set of $n$ points in cartesian coordinates(taken from marker on robot arm) and robot driver returns position of TCP(tool ...
1
vote
1answer
22 views

Negative definiteness of a block matrix

There is a block matrix, \begin{align} M=\left(\begin{array}{cc} A & B\\ C & I \end{array}\right)\quad\text{where}\quad A<0 \end{align} I am curious whether the matrix $M$ is negative ...
0
votes
0answers
20 views

Eigenvalues of integrals over similar matrices

Let $\rho = \rho(x)$ be a $2\times2$ matrix (don't know if it is necessary, but $\rho$ is a density operator) and $I$ be the (two-dimensional) identity matrix. I have two matrices $A$ and $B$, where ...
0
votes
1answer
15 views

Derivative of vector and vector transpose product

I saw this answer here : Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$. I am finding difficult to understand the part in red. What rule is that ? If I apply multiplication rule, ...
0
votes
0answers
16 views

Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? [duplicate]

Question: Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? Someone comes along and gives us the partition $P=\{2,2,3,3,4\}$ of $14$. How can we ...
0
votes
1answer
18 views

Characterising Adjugate(adjoint) of a matrix

If $A$ is an $n\times n$ matrix over a field, then adj$(A)$ is an $n\times n$ matrix (obtained from $A$) such that $$\mathrm{adj}(A)\,A=A\,\mathrm{adj}(A)=\mathrm{det}(A)I_n.$$ Question: If $B$ is ...
0
votes
1answer
50 views

Linear Algebra: Question about determinants

The following matrices are $4 \times 4$ matrices. $$A=\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1\\ 1 & 1& 1 &0\\ 1 &1 &0 &0 \end{bmatrix}\\ B= ...
0
votes
1answer
16 views

Sylvester's law of inertia for generic matrices.

By Sylvester's law of inertia, the positive and negative indices of a symmetric matrix $A$ are also the number of positive and negative eigenvalues of $A$. I was wondering if a similar result is known ...
0
votes
2answers
34 views

Eigenvalues of $\mathbb E\pmatrix{2X&X\\ 1-X&2X}$. [on hold]

Let $X$ be a random variable between $0$ and $1$, such that: $\mathbb{E}[X]=\frac{1}{2}$. We have a matrix: $$A=\left( \begin{array}{cc} 2X & X \\ 1-X & 2X \\ \end{array} \right)$$ ...
0
votes
3answers
42 views

Inverse of partitioned matrices [on hold]

A matrix of the form $$A=\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix}$$ is said to be block upper triangular. Assume that $A_{11}$ is $p \times p$, $A_{22}$ is $q \times q$ and ...
2
votes
0answers
57 views

Help me to prove the determinant formula

Actually it is about the question of n-linear function, but it is so relevant to the determinant formula. Here is the notation of the theorem. If $n>1$ and $A$ is an $n \times n$ matrix over $K$, ...
1
vote
1answer
33 views

Separating vectors from linear combination

Suppose I have a linear combination of vectors as follows $ \mathbf{s} = \alpha_1\mathbf{x}_1 + \dots + \alpha_m\mathbf{x}_m + \beta_1\mathbf{y}_1 + \dots + \beta_n\mathbf{y}_n $ where $\alpha_i, ...
2
votes
0answers
15 views

show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
2
votes
2answers
45 views

Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
2
votes
3answers
55 views

Producing lower bounds for $\text{trace}(A^2)$ for a positive semidefinite, symmetric matrix $A$

Are there any lower bounds on $\DeclareMathOperator{trace}{trace}$ \begin{align*} \trace(A^2), \end{align*} where $A$ is positive semi-definite and symmetric? I am aware of the inequality $$ ...
-1
votes
0answers
44 views

Find the solution to the following LPP by solving its dual. [on hold]

Minimize : $ Z = 300X_1 + 110X_2$ Subject to : \begin{align*} 30X_1 + 5X_2 &\geq 6 \\ 20X_1 + 10X_2 &\geq 8 \\ X_1, X_2 &\geq 0 \end{align*}