Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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57 views

Prove $Ax = \frac{1}{2}x$ only has the trivial solution where $A$ has all integer entries.

Prove $Ax = \frac{1}{2}x$ only has the trivial solution where $A$ is a $n \times n$ matrix with integer entries and $x = (x_1, \ldots , x_n)$. I am a bit rusty on my linear algebra and trying to ...
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1answer
25 views

Convert from one matrix to another

I have this 4x4 matrix A: a, b, c, 0, e, f, g, 0, i, j, k, 0, 0, 0, 0, 1 And I want to convert it to this matrix B: ...
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1answer
38 views

Set of normal forms for for $C_2\wr S_3$

I am working with the subgroup of $SL(3,\mathbb{Z})$ with generating set $$\omega=\begin{bmatrix} -1& 0& 0\\ 0& 1& 0\\ 0& 0&1 \end{bmatrix}r=\begin{bmatrix} 0& 1& 0\\ ...
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0answers
22 views

Alternative proofs invertible matrix/operator $T$ has square root over complex field using polar decomposition

My textbook gives the following proof: (i) $I+N$ where $N$ is nilpotent operator has square root (ii) $T$ may be decomposed into $T|_{G(\lambda_i)}=\lambda_i I+N_i$ (restriction to generalized ...
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0answers
41 views

Help with permutation lemma for PA=LU.

I am trying to prove that for any permutation matrix P. (I + c$E_{i,j}$)P = P(I + c$E_{si,sj}$) My approach so far: $P^{-1}$(I + c$E_{i,j}$)$P$ = ($P^{-1}$ + c$P^{-1}$ $E_{i,j}$)$P$ = I + c$P^{-1}$ ...
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2answers
44 views

Find the matrix of the orthogonal projector on span

I need to find the matrix of the orthogonal projector on $Span([1,1,-1],[1,-1,-1])$. I had used Gram-Schmidt process on it and have $Span([\frac{\sqrt{3}}{3},\frac{\sqrt{3}}{3}, -\frac{\sqrt{3}}{3}], [...
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0answers
59 views

Proof determinant and trace of transformation matrix of an endomorphism regarding different bases depend on the choice of the bases

I would be happy if someone could check this proof and tell me any mistakes I shall correct. Let $V$ be a finite-dimensional vector-space with $dim(V)>0$. Given an Endomorphism $g: V \to V$ we ...
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1answer
34 views

For which orthogonal $2\times2$ matrices does the exponential become orthogonal?

Let $ A$ be a $2\times2$ real orthogonal matrix. Then when does $e^A$ become orthogonal as well? According to my calculations, $A$ must be skew-symmetric also and therefore there are only two ...
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0answers
7 views

Matrix generator / representation matrix for equal spacing along a chirp function?

This question is similar to several sets of previous questions dating back to when I joined the site 1, 2, 3. This time it regards a sine function of quadratic frequency. a.k.a. a "chirp signal". I ...
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2answers
34 views

Find the vector $v$ such that there are an infinite number of solutions to the system of equations of the form $x^{*} + kv$

Q: The system of equations: $2x_{1}+x_{2}+3x_{3}=b_{1}$ $x_{2}+x_{3}=b_{2}$ $x_{1}+x_{3}=b_{3}$ b) Find the vector $v$ such that there are an infinite number of solutions to the system of ...
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1answer
52 views

What is the a matrix decomposition to swap out an eigenvalue?

I have the following problem: let $A$ be an $m \times m$ matrix that is not symmetric and real. Then, I am interested in its eigenspaces that correspond to the largest eigenvalue, $\lambda_{\text{max}}...
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1answer
25 views

How obtain the partial derivative of quadratic function $f(z)=\frac{1}{2} x^T(z)Q(z)x(z)$?

Consider $z\in \mathbb{R}$, $x(z):\mathbb{R} \to \mathbb{R}^{n\times 1}$ and $Q(z):\mathbb{R} \to \mathbb{R}^{n\times n}$ such that $Q(z)=Q^T(z)>0$. If we define the function $f(z):\mathbb{R} \to \...
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1answer
26 views

What is the complexity order of Kalman rank condition?

Does computing the Kalman rank condition of an integer matrix have complexity polynomial in the size of the input? if yes what is the order of complexity? For a discrete-time linear state-space ...
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0answers
57 views

Matrix method of solving non homogenous linear equations

For a real number $\alpha$, if the system of linear equations $\begin{bmatrix} 1 & \alpha & \alpha ^{2} \\ \alpha & 1 & \alpha \\ \alpha ^{2} & \alpha & 1 \end{bmatrix}$ $\...
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0answers
7 views

Clarification on theory behind finding: image, core, rank and defect of operator $f$

There is a linear operator $f:ℝ^3\to\mathcal{P}_3(x)$ which is defined by: $f(1,0,0)=1+x$, $f(0,1,0)=x+x^2$, $f(0,0,1)=2+x^2$ I need to find core, image, rank and defect of operator $f$. Results ...
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1answer
18 views

Decompose Frobenius norm of the difference between 2 matrices

Let $A\in\mathbb{R}^{n\times n}$ and $B = UDV^T$ where $U\in\mathbb{R}^{n\times r}$, $V\in\mathbb{R}^{n\times r}, D\in\mathbb{R}^{r\times r}$ and $U^TU = V^TV = I_r$. Define two sets: $$S_1=\mathbb{R}^...
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0answers
14 views

Cokernel of a matrix as a direct sum

Let $M$ be a $n \times m$ matrix with integer coefficients. Let $$\text{Coker}(M) = \mathbb{Z}^n /\, \text{Colspace}(M).$$ Why is it the case that there exist positive intergers $a_1, \cdots, a_{\...
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2answers
30 views

Operator norm of ABC where B is a PSD matrix

If B is a PSD matrix, I intuitively think it is true that $\|ABC\| \leq \|B\|\, \|AC\|$, but I can not prove it. Anybody can help? Thank you so much!
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1answer
31 views

relationship between rows and columns of an orthogonal matrix

Suppose $U$ is an $m\times n$ orthogonal matrix. Show that $m \geq n$. I'm having trouble with this proof -- I understand that the columns of $~U~$ can only be linearly independent in the cases ...
5
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1answer
80 views

Can $AB = \gamma BA$ for matrices $A$ and $B$

For what values of $\gamma \in \mathbb{C}$ do there exist non-singular matrices $A , B \in \mathbb{C}^{n \times n}$ such that $$AB = \gamma BA \,?$$ So far what I have done shown that $\gamma$ must ...
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1answer
22 views

Second method for finding matrix operator $f$ in base $\beta_2$

There is a linear operator $f:R^2 \rightarrow R^2$ with $f(x,y)=(4x-y, 2x+y)$ and there is base $\beta_2=\{(1,3)(2,5)\}$ in $R^2$. I need to find matrix of operator $f$ in base $\beta_2$. I got the ...
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2answers
43 views

Equation of a line after being reflected in another line using the transformation matrix

For a specific case, consider the line $ y = 3x + 1 $. How can I find the equation of the new line when this is reflected in the line $ y = 2x $ ? I would like to solve this using solely matrices and ...
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1answer
33 views

How to remove roll from axis angle rotation

I have an object in 3d space oriented along the global axes, with Z axis pointing forward, Y axis pointing up and X axis pointing right. I need to reorient this object to face a certain vector target <...
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1answer
46 views

Do we say that an $m \times n$ matrix A exists in $\mathbb{R}^m$ or $\mathbb{R}^n$?

As per the title, Do we say that an $m \times n$ matrix A exists in $\mathbb{R}^m$ or $\mathbb{R}^n$ ?
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1answer
25 views

Norm of a block of matrix operator

Let $(\mathcal{H}_1,\langle \cdot\mid \cdot\rangle_1), (\mathcal{H}_2,\langle \cdot\mid \cdot\rangle_2), \cdots, (\mathcal{H}_d,\langle \cdot\mid \cdot\rangle_d)$ be complex Hilbert spaces and let $\...
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0answers
39 views

Finding limit using Markov chain.

Here is the question: I set up the probability matrix as: \begin{bmatrix} 0 & 1/3 & 1/3 & 1/3 \\ 0 & 1/2 & 1/2 & 0 \\ 1 & 0 & 0 & 0 \\ 1/2 & 0 & 1/2 &...
1
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1answer
81 views

Proof of the matrix identity $\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B)$

The Wikipedia page about the determinant mentions the following matrix identity $$\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B),$$ valid for squared matrices $A$ and $B$ of the ...
3
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1answer
81 views

Rank-1 update eigenvalues

If I had a diagonal matrix with rank-1 update $$ D + uv^T $$ what can I say about its eigenvalues? I know from Two matrices that are not similar have (almost) same eigenvalues that every eigenvalue ...
1
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1answer
37 views

$\frac{1+m_v}{1+m_u}\leq \frac{1+u^T(M+I)^{-1} u}{1+v^T(M+I)^{-1}v} \leq \frac{1+m_u}{1+m_v}$ if $M$ is positive sym. PD & $u,v$ are $0-1$ vectors?

Let $n$ be a positive integer. Let $m_u,m_v \in \{1,...,n-1 \}$. Let $M$ be a $n \times n$ symmetric positive definite matrix with positive entries. Let $u$ and $v$ be vectors of length $n$ with ...
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0answers
24 views

Proof of equation involving fourth-order moments

How can you prove that $\mathrm{E}\left\lbrace \mathbf{x}(k) \mathbf{x}^{T}(k) \mathrm{E}\left\lbrace \Delta \mathbf{w} (k) \Delta \mathbf{w}^{T} (k)\right\rbrace \mathbf{x}(k) \mathbf{x}^{T}(k) \...
1
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2answers
48 views

How to solve $M - EV - (EV)^T = 0$ against $E$?

I have a matrix equation: $$M - EV - (EV)^T = 0$$ which i want to solve against $E$. Is this possible? How to do that? Remarks: matrices are square, $E$, $M$ are symmetric, $V$ is invertibile. ...
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3answers
56 views

How to find the set of solutions of $Ax=By$ for $A,B$ matrices?

Let $A,B$ be arbitrary matrices with the same number of rows. How can we find the set of solutions $x,y$ to the matrix equation $Ax=By$? I understand that this problem is probably related to that of ...
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1answer
22 views

Matrix calculations in second order polynomial approximation

I am looking at second order polynomial approximation, specifically at this link. However, I am stuck in the one dimensional case: I do not think I can calculate the following: $$ X^{T}*X^{-1}*X^{T}...
2
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0answers
62 views

Why are operators often written in 3s?

In a lot of my quantum mechanics and linear algebra books, operators are often defined as $M=IMI$ and many operations on operators are often done in similar fashion, for eg. operator $M$ might be ...
3
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1answer
89 views

Proof rank linear map is equal to the rank of its transformation matrix

I want to prove, that the rank of a linear map f: V $\rightarrow$ W is equal to the rank of the transformation matrix A of this linear map. Let $v$ a Basis of $V$ with length $n$ and $w$ a Basis of $...
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1answer
56 views

Is there an easy way to simplify $(P+Q)^2 - (P-Q)^2$ where $P$ and $Q$ are matrices?

P and Q are different matrices that are 2x2. I have tried to do this by letting both equal a 'made-up' matrix with pro numerals, but the expansion process is very long. Is there an easy way to ...
0
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2answers
71 views

When are the eigenvalues of a positive-definite matrix $\le 1$?

The eigenvalues of a positive-definite matrix are guaranteed to be $> 0$; but does anyone know of sufficient conditions when they will also all be $\le 1$?
5
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1answer
49 views

If $A\in {\mathbf F_2}^{n\times n}$ is symmetric, its diagonal is in the span of its columns

Let $A$ be an $n\times n$ symmetric matrix with entries in $\mathbf F_2$ (the field with 2 elements, also referred to as $\mathbb Z/2\mathbb Z$). Prove that the diagonal of $A$ is in the span of its ...
3
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1answer
80 views

Diagonalization of a symmetric matrix

I'm trying to prove that every symmetric matrix is diagonalizable. I know that there are already many answers regarding this question, but I just want to check out whether my approach is correct or ...
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0answers
11 views

Matrix perturbation and Eigenvector Bound

Recently I've been learning Matrix perturbation theory. Many theorems deal with perturbation bound in $l_2$ norm, or more generally, unitarily invariant norm, like F-norm or others(Like Davis-Kahan's ...
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3answers
37 views

similar matrices and one-to-oneness of matrix transformation

I need help with this proof Suppose $A$ and $B$ are $n \times n$ matrices and that there is an invertible matrix $P$ such that $A=PBP^{-1}$. Show that if $x \mapsto Bx$ is one-to-one, then $x \mapsto ...
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1answer
46 views

Showing $ u^T (M+I_n)^{-1} u \geq v^T(M+I_n)^{-1}v $ when $M$ is symmetric PD with positive entries and $u,v$ are $0-1$ vectors

Let $n$ be a positive integer. Let $m_u,m_v \in \{1,...,n-1 \}$. Let $M$ be a $n \times n$ symmetric positive definite matrix with positive entries. Let $u$ and $v$ be vectors of length $n$ with ...
1
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3answers
56 views

Showing $u^T M u \geq v^TMv$ when $M$ is symmetric PD and $u,v$ are $0-1$ vectors

Let $M$ be a $n \times n$ symmetric positive definite matrix. Let $u$ and $v$ be vectors of length $n$ with entries consisting $n-m_u$ (or $n-m_v$) $0$'s and $m_u $ (or $m_v$) $1$'s, where $m_u,m_v \...
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0answers
12 views

Why does Matrix Factorization (like ALS) lead to good results for missing values?

As the title already suggests, I'm not really understanding WHY Matrix Factorization methods like Alternating Least Squares (ALS) lead to good predictions of missing values. As far as I understand we ...
1
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1answer
26 views

Linear Algebra - Find an orthonormal basis

Question: Consider the space of $2 \times 2$ matrices equipped with the inner product: $\langle A , B \rangle = \operatorname{tr}(A^T B)$ i) Find an orthonormal basis for the subspace $V = \{M \...
1
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1answer
46 views

Proving that symmetric matrix is diagonalizable

I'm trying to prove that symmetric matrix (with real entries) is diagonalizable. Here, Ittay Weiss proved the result. I'm following with his argument, but I couldn't properly understand the following ...
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1answer
47 views

Prove that the spectrum of normal matrices is stable under small perturbations

Let $A$ be a normal matrix. As shown for example in these notes (link to pdf, see pagg. 120 and 121), given a matrix $B$ with a single non-zero entry and some parameter $\varepsilon\in\mathbb R$, the ...
2
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1answer
66 views

Finding the Gradient Matrix for the given expression

Let $\rho$ be a matrix, and let $\rho_A$ be the partial trace of the matrix $\rho$. For simplicity, let us assume $\rho$ is a $4 \times 4$ matrix. The partial trace is defined as follows: If $$\rho ...
2
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0answers
23 views

Calculating determinant of integral of matrices

I would like to calculate the determinant of some $3\times3$ matrix $$\boldsymbol{A}(t) = \int_{0}^{t}\boldsymbol{B}(s-t)\boldsymbol{C}(s-t)\boldsymbol{D}(s-t)ds$$ in terms of the determinants of the $...
4
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3answers
86 views

Is there a matrix that can be used to find the transpose of a matrix?

Let $A$ be a general $n\times n$ invertible matrix. Let $T^A$ be the "transposer" matrix i.e. $T^A A = A'$. (Does that $T^A$ multiplied by $A$ equal the transpose of $A$?) Then does $T^A$ depend on ...