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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23 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
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0answers
30 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}), ...
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1answer
47 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 ...
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1answer
243 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
64 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 ...
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1answer
39 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?
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4answers
1k 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 ...
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1answer
32 views

Matrix squaring identities

Prove that if $AB = BA$, then: $(AB)^2 = A^2 B^2 = B^2 A^2$ $(A + B)^2 = A^2 + 2AB + B^2$
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1answer
104 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
99 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 $i=n$. ...
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1answer
46 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 ...
1
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1answer
98 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
98 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 ...
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votes
2answers
50 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 $\det(P)=1$) ?
2
votes
2answers
73 views

is my working correct? Feedback appreciated!

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 $2x + 2y + z = 2$ $−x + 2y − z = −5$ ...
3
votes
1answer
83 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
81 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
115 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.
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2answers
57 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 $c \in C$
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votes
1answer
174 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 ...
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2answers
644 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
51 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
72 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 ...
2
votes
1answer
99 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
119 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 ...
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0answers
20 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
105 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
127 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
367 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 ...
3
votes
2answers
658 views

Why is the determinant of a rotation matrix equal to 1?

Why is the determinant of a rotation matrix equal to 1? I would like a geometric interpretation of this. Just curious.
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2answers
67 views

Find AB where A= matrix and B=matrix

$A=\left[\begin{array}{ccc} 2&1&0\\0&3&-1 \end{array}\right]$ $B=\left[\begin{array}{cc}a&1\\1&b\\b&a\end{array}\right]$ Matrices Find $AB$
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vote
1answer
430 views

Prove Solving a Lower Triangular Matrix By Forward Substitution is Backwards Stable

I'm taking a class in scientific computing and we are working on proving stability of certain algorithms. Unfortunately, at this stage, everything is proof-based, and I have little to no experience in ...
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votes
1answer
45 views

orthothogonal matrices [duplicate]

If a matrix $A$ is symmetric, i.e. $A=A^T$, can it also be orthogonal i.e. $AA^T=I$?
0
votes
1answer
78 views

Invariant subspaces of a linear operator

Can someone show me how to find all invariant subspaces of the real linear operator whose matrix is: $$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$ I have another example, so I would like ...
5
votes
0answers
45 views

Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
3
votes
2answers
96 views

Maximize the determinant

Over the class $S$ of symmetric $n$ by $n$ matrices such that the diagonal entries are +1 and off diagonals are between $-1$ and $+1$ (inclusive/exclusive), is $$\max_{A \in S} \det A = \det(I_n)$$ ...
5
votes
1answer
230 views

If $A,B,C$ are nonsingular, so is $A\sin(t)+B\cos(t)+C$ for some real $t$

While trying to answer another question on this site, I found that I needed the following assertion: If $A,B,C$ are nonsingular complex matrices of the same sizes, then $A\sin(t)+B\cos(t)+C$ is ...
1
vote
2answers
408 views

Finding the determinant of a $4\times4$ matrix

How does one find the determinant of a $4\times 4$ matrix? I am using Cramer's rule to solve a system of linear equations but don't know how to find the determinant of a $4\times 4$ matrix. Our matrix ...
0
votes
3answers
77 views

Basis of a 2 dimentional vector space

Let $T: V \rightarrow V$ be a linear operator on a vector space of dimension 2. Assume that $T$ is not multiplication by a scalar. Prove that there is a vector $v$ in $V$ such that $(v,T(v))$ is a ...
4
votes
1answer
211 views

derivative of log(det(A)) wrt x, where A is matrix that depends on x

I have two large sparse matrices B and C, and I need to calculate $\frac{\rm{d}}{\rm{d}(\log({\lambda}) }\log( \det(B+\lambda C))$. Because B and C are very large I can't directly evaluate the ...
1
vote
1answer
73 views

Conjugation by elementary matrices

Let $ A=\begin{pmatrix} a & b \\ c & d \end{pmatrix} $ be a real matrix, with $c$ not zero. Show that using conjugation by elementary matrices, one can "eliminate" the $a$ entry.
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1answer
50 views

On determinants computation

How can be proved this identity between determinants? $$\left|\begin{array}{cccc} 1&a&c&ac\\ 1&b&c&bc\\ 1&a&d&ad\\ 1&b&d&bd \end{array}\right|=\left| ...
0
votes
1answer
360 views

Consistency of a System of linear equations

Test the consistency of the system of linear equations $$\begin{align} 4x-5y+z & =2 \\ 3x+y-2z& = 9 \\ x+4y+z& =5\end{align}$$
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0answers
20 views

How to set dihedral values to null?

I have a protein with many residues, but I would like to set the phi and psi angles of residue 15 to value of null. I have a file containing all residues and Cartesian coordinates, and I have another ...
4
votes
2answers
91 views

Determinants of Block Matricies

I read on wikipedia that $Det \begin{pmatrix} A & B\\B& A\end{pmatrix}$ is equal to $ Det(A+B)Det(A-B) $ if $A$ and $B$ commute. Does this hold true even if $ A $ and $ B$ are not ...
1
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0answers
34 views

Is there a special name for matrices with $M[j,i] = M[i,i] - k, i \neq j$?

Backgroud: I am working on a computer science problem and arrives at a matrix $M$ with the following property: The size of Matrix $M$ is $n\times n$. For each row $j$, we have $M[j,i] = ...
2
votes
0answers
51 views

Matrix of integers to boolean matrix

My Question is about converting a matrix of numbers, say each row is an item and each column is a feature of the item. The features are currently integers but I want to convert the feature ...
0
votes
1answer
87 views

Clarification on matrix notation subscript and superscript notation

If a matrix C exists in integers $\mathcal{Z}_q^{mxl}$ what does this mean?
0
votes
2answers
86 views

What is determinant? [duplicate]

I know this can be the most stupid question here. However, what I want to ask is not how to compute determinant or what the definition of determinant is.(Enough homework :P ) What I really want to ...
1
vote
1answer
81 views

homework - Find a basis for the space of all vectors in R6 with x1 + x2 = x3+ x4 = x5+ x6

a) Find a basis for the space of all vectors in $\mathbb{R}^6 $ with $x_1 + x_2 = x_3 + x_4 = x_5 + x_6$. b) Find a matrix with that subspace as null space. c) Find a matrix with that subspace as ...