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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2
votes
1answer
218 views

Eigenvalues of a matrix $A$ and corresponding linear map (Linear algebra: Hoffman kunze 6.2.15)

Let $V$ be the vector space of $n\times n$ matrices over the field $F$. Let $A$ be a fixed $n\times n$ matrix over $F$. let $T$ be a linear operator on $V$ defined as $T(B) = AB$. Question is to ...
5
votes
4answers
352 views

if $AB\neq 0$ for any non zero matrix $B$ then $A$ is invertible

Question is to check that : If $A$ is an $n\times n$ matrix over a field $F$ and $AB\neq 0$ for any non zero matrix $B_{n\times n}$ over $F$ then, $A$ is invertible. This does make some sense to me ...
1
vote
1answer
247 views

Computing the number of positive and negative eigenvalues

Given a $n \times n$ symmetric matrix $A$ with integers as entries I would like to compute the number of strictly negative $\rm{nn}(A)$ and positive $\rm{np}(A)$ eigenvalues of $A.$ My question is ...
6
votes
2answers
9k views

Distance/Similarity between two matrices

I'm in the process of writing an application which identifies the closest matrix from a set of square matrices $M$ to a given square matrix $A$. The closest can be defined as the most similar. I ...
1
vote
0answers
231 views

Matlab (machine learning) - How to calculate binary scatter plot with straight line distinguishing these values?

Maybe title sounds a bit confusing, so I'll try to explain it. Let say we have a some matrix 4x4 = (0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0) So some function should ...
7
votes
4answers
228 views

General question about matrix calculus with specific example (with attempted answer)

I'm struggling to find the right way to approach matrix calculus problems generally. As an example of a problem that is bothering me, I would like to calculate the derivative of $||Ax||$ (Euclidean ...
22
votes
6answers
18k views

Show that the determinant of $A$ is equal to the product of its eigenvalues.

Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I'm having a tough time figuring this one out. I know that I have to work with the characteristic ...
6
votes
5answers
3k views

Intuition behind Matrix being invertible iff determinant is non-zero

I have been wondering about this question since I was in school. How can one number tell so much about the whole matrix being invertible or not? I know the proof of this statement now. But I would ...
3
votes
4answers
282 views

minimal polynomial of a matrix with some unknown entries

Question is to prove that : characteristic and minimal polynomial of $ \left( \begin{array}{cccc} 0 & 0 & c \\ 1 & 0 & b \\ 0 & 1 & a \end{array} \right) $ is ...
1
vote
1answer
304 views

Rank of matrices that are the products of square and non-square matrices

Can someone give me a proof for the following statement? $B$ is an invertible $n\times n$ matrix, then the rank of $AB$ is the same as the rank of $A$ for every $m\times n$ matrix $A$. Is the ...
0
votes
2answers
1k views

Invertibility and Rank of matix

Can anyone give me a proof for, B is an invertible $n$x$n$ matrix, then the rank of $AB$ is the same as the rank of $A$ for every $m$x$n$ matrix $A$. Also, is the converse true for the statement ...
4
votes
1answer
1k views

Singular values of square orthogonal matrix?

What are the singular values of an $n \times n$ square orthogonal matrix? How do we know that the set of all orthogonal matrices is convex? Is there an example?
1
vote
1answer
122 views

invertibility of non square matrix

I was looking at the properties of invertible matrix, and I came across this statement. If $A$ is $m$ by $n$ matrix and the $rank$ of $A$ is equal to $m$, then $A$ has a right inverse. Can any give ...
1
vote
1answer
3k views

Finding determinant by applying Gaussian Elimination

(I don't know how to make a matrix here, someone please correct it into a better format, thanks~) So I'm applying the Gaussian Elimination to find the determinant for this matrix: $\begin{pmatrix} ...
1
vote
1answer
90 views

Sharp bound on off-diagonal entries of matrix with 1's on diagonal to make matrix invertible

Suppose $A$ is an $n \times n$ matrix with all 1 on the diagonal. What is the sharp bound $\epsilon(n)$ so that $A$ is invertible if all off-diagonal entries of $A$ have absolute value less than ...
2
votes
1answer
114 views

Homework - ERO with Unknown in Matrix

I was having a problem with how to properly perform elementary row operation (ERO) on a matrix. In the question, we were given an augmented matrix ...
3
votes
2answers
175 views

$A$ is invertible if and only if $A^t$ is invertible

I hate these "easy" proofs. They always slip under my radar. How do I show that a square matrix $A$ is invertible if and only if $A^t$ is invertible?
4
votes
0answers
89 views

The determinant of a special matrix

Recently, I encounter the problem of calculating the determinant of the following matrix $$\left(\begin{array}{cccc} \sin(\theta_1) & \sin(\theta_1 + \delta_1) & \cdots & \sin(\theta_1 + ...
1
vote
2answers
164 views

Possible ranks of a matrix

Let $v=(a_1,\cdots,a_n)$ be a real row vector. We may form the $n! \times n$ matrix $M$ whose rows are obtained by permuting the entiers of $v$ in all possible ways. The rows can be listed in an ...
3
votes
3answers
305 views

Expressing the trace of $A^2$ by trace of $A$

Let $A$ be a a square matrix. Is it possible to express $\operatorname{trace}(A^2)$ by means of $\operatorname{trace}(A)$ ? or at least something close?
0
votes
2answers
1k views

Find a matrix so that $A^2$ not equal to 0 but $A^3$ is [Strang P78 2.4.23]

(a) Find a nonzero matrix $A$ for which $A^2 = 0$. (b) Find a matrix that has $A^2 \neq 0$ but $A^3 = 0$. Solution for (a): Let $A := \text{column $\times$ row} = \mathbf{cr^T} \neq \mathbf{0}$ ...
0
votes
1answer
53 views

Differences of skew symmetric matrices

Let $A$ be an invertible real skew-symmetric matrix, and consider the difference $A_R:=RAR^{-1}-A$, for orthogonal $R$. Is it true that $A_R$ is either zero or invertible? Does the answer depend on ...
1
vote
0answers
38 views

kernel of linear application

Let $n \in \mathbb{N}^*$, $A \in \mathbb{R}^{n}$, $A \neq 0$ and $\Phi_{A} \, : \, \mathcal{M}_{n}(\mathbb{R}) \, \rightarrow \, \mathbb{R}$ such that : $\forall M \in \mathcal{M}_{n}(\mathbb{R}), ...
0
votes
1answer
58 views

How to Differentiate this Matrix product

I am trying to solve the matrix equations for linear regression and it leads me to the following differentiation. I cannot find an explanation on how to do it on the Internet, only the result being ...
-1
votes
1answer
377 views

ker(AB) = ker(A) + ker(B)

I'm trying to prove the following: Let $A$ and $B$ be two commutative square matrices ($AB=BA$) over a commutative field such that $Im(A)=ker(A)$ and $Im(B)=ker(B)$. Then $ker(AB) = ker(A) + ker(B)$. ...
4
votes
2answers
75 views

What have Vectors and Matrices got to do with each other?

In my undergraduate course work I learnt Vectors (as in those in vector space with magnitude and direction) separately from Matrices - an $n \times m$ array of numbers. However, after sitting in for a ...
0
votes
1answer
49 views

How do you solve invertible matrices?

Prove the property: If A is invertible and k does not equal 0 number, then kA is invertible and (kA)-1(inverse) =(1/k)A-1(inverse) Can k equal 1 and then solve this?
-1
votes
4answers
5k views

How to prove $A+ A^T$ symmetric, $A-A^T$ skew-symmetric.

Prove that if $A$ is a square matrix, then: a) $A+ A^T$ is symmetric. b) $A-A^T$ is skew-symmetric. c) Use part (a) and (b) to show $A$ can be written as the sum of a symmetric matrix $B$ and a ...
0
votes
1answer
197 views

What is $\rho$ and $\sigma$ in this theorem?

This might be a silly question but, heres a note I made in linear algebra class: Suppose we have $Ax = \lambda x$, then $\rho(A)x = \rho(\lambda)x $, so $\rho(\sigma(A)) \subset \sigma(\rho(A))$. My ...
4
votes
3answers
264 views

Proof with binomial coefficient and kronecker delta

I want to prove that $$ \sum_{k=i}^n \binom{n}{k}\binom{k}{i}(-1)^{n-k}=\delta_{n,i} $$ Where $\delta_{n,i}$ is the Kronecker Delta, i.e. $\delta_{n,i}=0$ if $n \neq i$ and $\delta_{n,i}=1$ if $i=n$. ...
0
votes
1answer
67 views

Condition Number of Polynomial (Condition Number = 0)

I'm calculating the condition number of a polynomial equation $$ y = (x-2)^{9} $$ for this equation, the Jacobian is equal to ...
3
votes
1answer
344 views

Finding matrix norm equivalence constants

I've been given the following: "Find the best positive constants $\alpha$ and $\beta$ such that $\alpha\left\|A\right\|_2\leq\left\|A\right\|_1\leq\beta\left\|A\right\|_2$ for all ...
3
votes
1answer
113 views

Diagonalizing using a matrix $P$

Let $A=\begin{pmatrix} a & b \\ c& d \end{pmatrix}$ be a $2 \times 2$ matrix witth eigenvalue $\lambda$. (a) Show that unless it is zero, the vector $\begin{pmatrix} b \\ \lambda -a ...
0
votes
2answers
79 views

Diagonalization with a matrix in $SL_n(\mathbb{R})$

Suppose that $A$ is diagonalizable. Can the diagonalization be done with a matrix $P$ in the special linear group $SL_n(\mathbb{R})$ (i.e. such that $\det(P)=1$) ?
3
votes
3answers
132 views

Solving a system of three linear equations with three unknowns

Is my working correct or am I completely wrong? Have I missed anything out? Any feedback is appreciated. Question: Consider the following system of equations $2x + 2y + z = 2$ $−x + 2y − z = −5$ ...
4
votes
1answer
97 views

Easy proof that $\exp{Xt} = I \Rightarrow X = 0$

Let $X\in \mathbb{C}^{n\times n}$ and $I$ is identity matrix , than if: $$ \forall t\in \mathbb{R}\quad e^{Xt} = I $$ than $$ X = 0. $$ I'm looking for short and slick proof of this ...
4
votes
1answer
88 views

Simple question about matrices

My question is simple : If one replaces some of the entries of a matrix by 0, does he obtain necessarily a matrix with a lower norm? I have to precise that the norm I use is the maximum of the ...
0
votes
2answers
131 views

Why determinant map matrices is a polynomial and not identically zero?

Let $A,B \in M_n(C)$ are invertible then we consider the map $c \rightarrow det(A+cB)$ which is a polynomial. How to prove that the polynomial $det(A+cB)$ not identically zero? thanks in advanced.
1
vote
2answers
58 views

$P+cQ$ is invertible for a finite number

Since $C$ is a field and $P,Q \in M_n(C)$ are invertible, can any body show me that $P+cQ$ is invertible for all but a finite number $c \in C$
0
votes
1answer
301 views

Sylvester's criterion about positive definite matrices.

The below quote is copy from "Problems and Theorems in Linear Algebra" Author is : V.Prasolov Let $A=||a_{ij}||_{1}^{n}$ be an Hermitian matrix, if $A$ is positive definite, then the matrix ...
0
votes
2answers
1k views

Prove that the determinant of a householder matrix is -1

I understand that a householder matrix has eigenvalues of either 1 or -1, however I isn't clear to me why the determinant is -1. Clearly the determinant is equal to the product of the eigenvalues so ...
0
votes
1answer
95 views

Recursive relation for a characteristic polynomial

I need to find a recursive relation for the characteristic polynomial of the $k \times k $ matrix $$\begin{pmatrix} 0 & 1 \\ 1 & 0 & 1 \\ \mbox{ } & 1 & . & . \\ \mbox{ } ...
3
votes
1answer
146 views

Characteristic Polynomial of $A$ and polynomials annihilating $A$

If $A$ is a real $3 \times 3$ matrix which is not diagonal. $p$ is a polynomial of degree 3 with real coefficients which is annihilating $A$. I have proved that if $A$ has a complex root (with non ...
0
votes
1answer
135 views

What does the notation $H\biguplus RH$ mean?

I have some problems understanding the notation used in this question. Let $K:= \left\{P\in GL_{2}\mathbb{(R)}: P^{T}P=I_{2}\right\}, H:=\left\{A_{\theta}=\begin{pmatrix} \cos(\theta) & ...
2
votes
1answer
114 views

Compute $e^{tA}$

When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(. Here's my problem: For the system of equations: $$\begin{cases} & \text{ } ...
3
votes
1answer
132 views

Idempotents in $M_2(\mathbb{C})$

Given two idempotents $e,f\in M_2(\mathbb{C})\setminus\{I_2\}$, the sets $$\{eg^{-1}:g\in GL_2(\mathbb{C}), eg^{-1} \text{ is an idempotent}\}$$ and $$\{gf:g\in GL_2(\mathbb{C}), gf \text{ is an ...
1
vote
0answers
24 views

Compute $\|A(t)\|$

When I do my homework (stability theory), I must use the knowledge to the norm of matrix. But I don't remember it (I mean that I'm not sure). My problem: For matrix $A(t)$ is continuous. Compute ...
1
vote
0answers
122 views

A Rank-One Reduction Formula

Consider $A_{m\times n}$ is very large, dense and full rank matrix. How can I find matrix B such that $\operatorname{rank}(B)=\operatorname{rank}(A)-1$? (Rank reduction formula must be invertible and ...
1
vote
2answers
130 views

What is this question asking me? Help appreciated!

I was wondering if anyone would be able to help me understand what this question is asking me. How would I go about working these out on Wolfram Alpha? I'm not too sure how to input them. Any help is ...
0
votes
3answers
837 views

Rotation of matrices

I am doing rotation of matrices at the moment, I know that if I want to rotate a point, let's say (2,1) 90 degrees clockwise, I have to multiply the matrix [ 2 1 ] * [0 1, -1 0] , but how do I find ...