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
4answers
163 views

Proving that $\operatorname{rank}(AB)$ is smaller or equal to $\operatorname{rank}(B)$ [duplicate]

I am struggling with proving the theorem that if $A$ and $B$ are $n\times n$ matrices, then: $$\operatorname{rank}(AB)\leq \operatorname{rank}(B)$$ Could anyone suggest me a hint? Any help is ...
2
votes
1answer
901 views

Priority vector and eigenvectors - AHP method

I'm reading about the AHP method (Analytic Hierarchy Process). On page 2 of this document, it says: Given the priorities of the alternatives and given the matrix of preferences for each ...
2
votes
1answer
86 views

Central extension of the Discrete Heisenberg group $H_3(\Bbb Z)$

I want to use the Discrete Heisenberg group $(H_3(\Bbb Z),\times)$ as an example for a presentation on central extensions. $H_3(\Bbb Z) = \begin{bmatrix}1&x&z\\0&1&y\\0&0&1 ...
2
votes
1answer
89 views

Eigen value of principal submatrix.

I was studying "interlacing property" and trying to find out the below fact- $A$ is an adjacency matrix of a $r$ regular graph $G$. $u,v \in G $;$u,v$ are not similar vertices. $B$ is the ...
2
votes
1answer
52 views

How can I prove that any matrix A can be expressed as the sum of two Hermitian matrices , B and C, in the form A = B + iC?

The question is in the title really. Whether or not A must also be Hermitian is not clear to me. Sorry, I am not very good with proofs of this nature.
2
votes
1answer
150 views

If $A$ is positive definite then so is $A^k$

I know how to show the inverse of positive definite is positive definite but I don't know how to expand that. Suppose $A$ is positive definite then $A$ is invertible, so define $y=Ax$ for $x\neq 0$. ...
2
votes
3answers
62 views

How does this reduced matrix indicate that the vectors are linearly independent?

I know that a set of vectors is linearly independent when a linear combination of them equal to zero is only satisfied by coefficients that are all zero. For this particular question, we have a ...
2
votes
1answer
92 views

Is there an easier way to find the inverse of a 3x3 matrix?

I know the normal process is to do row operations to transform the matrix to get the identity matrix and then apply the same row operations in the identity matrix to get the inverse. But this process ...
2
votes
1answer
599 views

The number of linearly independent solution of the homogeneous system of linear equations $AX=0$

I came across the following multiple choice question: The number of linearly independent solution of the homogeneous system of linear equations $AX=0$, where $X$ consists of $n$ unknowns and $A$ ...
2
votes
3answers
189 views

Exponential of a matrix always converges

I am trying to show that the exponential of a matrix converges for any given square matrix of size $n\times n$: $M\mapsto e^M$ e.g. $\displaystyle e^M = \sum_{n=0}^\infty \frac{M^n}{n!}$ Can I argue ...
2
votes
1answer
41 views

Dimension of set of Hermitian matrices commuting with a given matrix

Given a Hermitian matrix $A$, what is the dimension of the set of all other Hermitian matrices $B$ such that $[A,B] = 0$. It is clearly not the same for all $A$, but how can one find it for a given ...
2
votes
1answer
59 views

Matrix inequality: Eigenvalues

A is a $n \times n$ non symmetric matrix. Can $\sigma_1(A)$ - the greatest singular value of A - be upper bounded by some function of the eigenvalues of A?
2
votes
1answer
32 views

Inequality $|A+B|_m\leq|A|_m+|B|_m$ on square matrices

Consider $n\times n$ real matrices $A$ and $B$. If $|A|_m$ denotes the modulus matrix of $A=[a_{i,j}]_{n\times n}$, and is defined as $|A|_m := [|a_{i,j}|]_{n\times n}$, prove that ...
2
votes
3answers
39 views

Why is the matrix [1,0;0,0] not positive definite?

If I take a vector v = [a,b], then isn't v.Mv = a^2, which is strictly greater than zero for all a and b not equal to zero?
2
votes
2answers
87 views

Inverse of a special matrix

Is there easy (analytical) way to find the inverse of the following matrix, where $C$ is a vector? $$ \begin{bmatrix} 1 & C^\top \\ C & CC^\top \end{bmatrix} $$
2
votes
1answer
620 views

A matrix is an orthogonal projection if idempotent and symmetric.

I have a matrix $A=\mathbf{v}\mathbf{v}^t$ where v is a vector in $\mathbb{R}^n$ with magnitude $1$. I have to prove that $A$ represents an orthogonal projection onto span$\{\mathbf{v}\}$. I have ...
2
votes
1answer
112 views

What is the purpose of finding the kernel of a matrix

Can someone help me understand why you would ever want to calculate the kernel of a matrix? I kept on trying to find applications when $\ker(A) = 0$ is useful but I could not find any. Can someone ...
2
votes
1answer
209 views

Let A be a symmetric positive definite matrix. Find a matrix B such that $B^2=A$

I believe this question is the same as asking find matrix B to be the square root of the matrix A. $B=\sqrt A$. Since the problem is not specific I am thinking to solve it in the general case by ...
2
votes
1answer
94 views

If $A$ is a square matrix that is linearly independent, is $AA$?

I'm just not sure how to start this problem from Linear Algebra Done Wrong. The problem is to prove that if the columns of $A$, square matrix, are linearly independent, then the columns of $A^2$ = ...
2
votes
2answers
148 views

Why use homogeneous coordinates?

I am having trouble understand the use of homogeneous coordinates for when describing transformations in 3D space. From what I have seen, the only difference between a transformation matrix in ...
2
votes
1answer
339 views

Under what conditions is $AA^T$ invertible?

Given a matrix $A$ with dimensions $m \times n$, is $B=AA^T$ invertible if and only if the rows of $A$ are linearly independent? So far, I've tried writing A as row vectors, $$A = ...
2
votes
1answer
58 views

Linear algebra matrix inverse identity

Consider an $m\times n$ matrix $A$ which is full rank. Is $A(A^\top A)^{-1}A^\top= I$ where $I$ is the identity matrix? If so how can this be shown? Note: it may be assumed that the matrix $A$ has ...
2
votes
2answers
70 views

Representing a $2 \times 2$ matrix as a $1 \times 4$ vector?

It seems to me (acording to assingment solutions), that you can write a $2 \times 2$ matrix as a column vector instead. Why can you do that? I just saw a solution to an assignment involving ...
2
votes
2answers
96 views

Nullity and Rank without a clear matrix

For a fixed non-zero vector b =$\left( \begin{array}{c} b_1 \\ b_2 \\ b_3\\ \end{array} \right)$ , the mapping $\;T : \mathbb R^{3}\ \to\ \mathbb R^{3}\\;$ is defined by $T$(x) = x ...
2
votes
1answer
33 views

The characteristic polynomial of A matrix A^k related to A

when I'm learning markov matrix markov pdf download here when proving THEOREM 4.12, It said, characteristic polynomial of a matrix A $ch(A) =(x−c_1)^{a_1} ···(x−c_t)^{a_t} ⇒ ch(A^k) =(x−c_1^k)^{a_1} ...
2
votes
1answer
180 views

Finding the minimal polynomial from characteristic equation

I am attempting to find the minimal polynomial of a matrix. My characteristic equation turns out to be $x^3 - x$ which factors out to $x(x-1)(x+1)$. Now, I am reading that the minimal polynomial is ...
2
votes
1answer
60 views

The determinant of a matrix exponential?

So I know that if a matrix $A = e^{B}$, then $det(A) = e^{tr(B)}$. I'm wondering, is the converse true? Right now I have a matrix and I know its determinant is $e^{tr(B)}$. So can I conclude that my ...
2
votes
3answers
84 views

Should a reflection matrix of a vector have the same form as a rotation matrix?

According to the book: I know that it is not possible to write a reflection as a rotation, but from the text it seems that the matrix of the form $$ A=\left[ \begin{array}{ c c } a & ...
2
votes
1answer
91 views

Commutativity of the square root of matrices

Let $A, B \in \mathbb{R}^{n \times n}$ two positive definite matrices such that $AB = BA$, that is $A$ commutes with $B$. It is easy to prove that $A^{1/2}$ commutes with $A$, indeed $AA^{1/2} = ...
2
votes
1answer
72 views

determinant of matrix $X$

Please hint me. ‎How ‎can I ‎calculate ‎determinant ‎of ‎matrix ‎‎$‎X‎$‎?‎ \begin{equation*}‎ ‎\mathbf{X}=\left(‎ \begin{array}{ccc}‎ A&B&‎\cdots&B\\‎ B&A&‎\cdots& B\\‎ \vdots ...
2
votes
1answer
70 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
2
votes
1answer
65 views

Example of a non singular square matrix such that $A+A^{-1} = 0$

Is there any example of a non singular square matrix $A$ such that $A+A^{-1} = 0$? Are they any specific type of matrices or can these be found under any category of matrices (such as symmetric, ...
2
votes
2answers
77 views

Why is the similar of a triangular matrix unipotent

If $ A = BDB^{-1} $, $B \in Gl_n(K)$ and $ D = (d_{ij}) $ an upper triangular matrix with 1 on the diagonal line. Show that A is unipotent, using the definition that a matrix A is unipotent if there ...
2
votes
1answer
80 views

Can someone direct me to prove the following three claims about a rotation group $SO(3)$ and its Lie Algebra?

In class my prof made three claims about a group and its Lie algebra. I cannot find direct reference to these claims because they are delivered in verbatim (im not even sure if I have them jogged down ...
2
votes
2answers
76 views

Singular matrix geometric sum

What is a fast way to calculate the sum $M + M^2+M^3+M^4+\cdots+M^n$, where $M$ is an $n \times n$ matrix whose cells are either $0$ or $1$? I have researched an alternative way which makes use of ...
2
votes
1answer
22 views

Multiplying matrices to get a specific result

Are there matrices $A,B$ (of dimension $n$), that give \begin{equation} AB-BA=I \end{equation} I have tried getting a result in small scale by using $2\times 2$ matrices and got a false equation ...
2
votes
2answers
52 views

How to decompose this matrix exponential?

I would some help with the steps to decompose the below matrix exponential. $\exp\left[ \zeta \left ( \begin{matrix} -\cos(x) & i \sin(x) \\ -i \sin(x) & \cos(x) \end{matrix} \right ) ...
2
votes
2answers
30 views

Must basis for column space be consist columns?

Lets say we have the following Matrix $$\left[ \begin{matrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 0 \\ \end{matrix} \right]$$ Obviously a basis for this would be ...
2
votes
1answer
190 views

identity of $(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+…+(I-w^{n-1}zT)^{-1}]$

I am trying to understand the identity $$(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+...+(I-w^{n-1}zT)^{-1}] \quad (*),$$ where $T \in \mathbb{C}^{n\times n},z\in \mathbb{C}$ and the ...
2
votes
1answer
58 views

If $X=PX'$, then P is invertible?

Let V be a set of all $n \times 1$ column vectors and $X,X'$ is in V. If there exist $n\times n$ matrix P such that $$X=PX'$$ then can we say P is invertible? I guess that $X=0 \Leftrightarrow ...
2
votes
1answer
64 views

$n\times n$ matrix with all eigenvalues equal to $1$ or $0$. Does a conjugated matrix with only $1$'s and $0$'s exist?

Let $A$ be an $n\times n$ matrix with all eigenvalues equal to $1$ or $0$. Is there a conjugated matrix $B = XAX^{-1}$ for some $X$ such that all the elements equal either $1$ or $0$? My thoughts so ...
2
votes
2answers
69 views

Jordan form of a matrix

Let $$A = \left( {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & 2 & 0 \cr 0 & 0 & 0 & 3 \cr 0 & 0 & 0 & 0 \cr } } \right)$$ The ...
2
votes
1answer
63 views

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. If $A$ is real, show that every eigenvalue of $A$ is real.

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. (a) If $A$ is real, show that every eigenvalue of $A$ is real. (b) If $A \in M_n$ has real main diagonal entries and its ...
2
votes
1answer
98 views

Relationship between similarity and having the same minimal polynomial

Let $A$, $B$ $\in M_3$ be nilpotent, where $M_3$ is the set of all complex 3by3 matrices. Show that $A$ and $B$ are similar if and only if $A$ and $B$ have the same minimal polynomial. Is this true in ...
2
votes
1answer
51 views

Conditions for invertibility of $AA^t$

Let $A$ be a matrix whose rows are pmfs (i.e. nonnegative entries, each row sums to $1$). Are there any conditions on $A$ weaker than invertibility such that $AA^t$ is invertible?
2
votes
1answer
150 views

Jacobson radical of a certain ring of matrices

Given a Matrix $A \subset M_4(\mathbb{C})$ be the $\mathbb{C}$-subalgebra consisting elements in the form \begin{pmatrix} * & * & * & *\\ * & * & * &*\\ 0 & 0 & ...
2
votes
1answer
88 views

How do I find the characteristic polynomial and eigenvalues?

For the following matrix, compute its characteristic polynomial its eigenvalues $$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$ So I think I ...
2
votes
2answers
45 views

Find all 2 x 2 skew-symmetric matrices A [closed]

Find all $2 \times 2$ skew-symmetric matrices $A$, if any, such that $A^2 + I_2 = 0$ Please help me! Thanks!!
2
votes
1answer
50 views

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$ Having no success with this question, I turn for your help =] I ...
2
votes
1answer
360 views

Jacobian Matrix in dynamical systems

Can someone explain what exactly the Jacobian matrix is (specifically in its application to dynamical systems) and maybe give an example of how to compute it? It really confuses me...and I haven't ...