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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Can someone help me, to compute by hand $Ax=b$ where $A$ is a $5\times5$ Matrix and $b=(1,1,1,1,1)$ transposed

Using Cholesky decomposition i have L and L transposed,how can i solve $Ax=b$ without using inverses? $A$ is a $5\times 5$ Matrix $b=(1,1,1,1,1)^t$ Another question, How would you easily compute the ...
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2 votes
1 answer
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Proof of Wronskian relation using induction

We have the following linear homogenous DE system $X' = AX, \tag 0$ I wanna prove with induction that $dW/dx = Tr(A)*W$ So for n=2 based on the above, we get, $A = \begin{bmatrix} a_{11} & a_{12} \...
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3 votes
0 answers
14 views

Divisibility of unitriangular matrices over a field of characteristic 0 [duplicate]

Definition: A group $G$ is said to be divisible if for any nonzero integer $n$ and for any $g \in G$ there exists $h \in G$ such that $g = h^n$. Let $U_n(k)$ be the subgroup of unitriangular $n \times ...
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What set of matrices $X,Y$ satisfy $X \circ Y = X P Y$ for at least one permutation matrix $P$?

Suppose we have three $n \times n$ matrices $X,Y,P$ where $X,Y \in \mathbb{R^{n \times n}}$ and $P$ is a permutation matrix. Let us take $\circ$ to be the element-wise product between two matrices. ...
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1 vote
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Hyperplane contains an inversible matrix

Let $H$ be an hyperplane of $M_n(\Bbb K)$ $(n\ge 2)$ Show that there exists $A \in M_n(\Bbb K)$ such that $H=\{M\mid \text{tr}(AM)=0\}$ Deduce that $H$ contains an invertible matrix
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1 vote
3 answers
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Let $A\in\mathbb{R}^{2\times 2}$ be such that $\det(A)=d\ne0$ and $\det(A+d\cdot\text{Adj}(A))=0$. Evaluate $\det(A-d\cdot\text{Adj}(A))$.

Let $A$ be a $2\times2$ matrix with real entries such that $\det(A)=d\ne0$ and $\det(A+d\cdot\text{Adj}(A))=0$. Evaluate $\det(A-d\cdot\text{Adj}(A))$. My Attempt I am multiplying the matrices. $(A+d\...
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3 votes
1 answer
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Given matrix associated to linear transformation find the basis it corresponds to

I know how to find the matrix associated to a linear transformation with respect to two given basis, but how can one find the basis given the matrix? Let's say we're given a linear transformation $T : ...
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-2 votes
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Find Matrix A such that $\operatorname*{argmin}_A ||H-A||^2_F + |A|_1 $ from given matrix $H$

Question I have matrix $H$, I want to find Matrix $A$ such that: $$ \operatorname*{argmin}_A ||H-A||^2_F + |A|_1 $$ How can I do that? What's the rule of updating? Can someone please guide?
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3 votes
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When does a symmetric matrix admit a positive eigenvector (i.e. $v_i >0 \forall i$) for a positive eigen?

Let $S \in \mathbb{R}^{n \times n}$ be a symmetric matrix. I am trying to understand when such a matrix possesses a positive eigenvector for a positive eigenvalue, that is a $v \in \mathbb{R}^n, v_i &...
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2 votes
0 answers
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Approximating the eigenvalues and eigenvectors of $B := A + E$

Suppose I have a matrix $B := A + E$, where $A$ is diagonal and $E$ is an off-diagonal, symmetric matrix whose non-diagonal elements are small. Is there any way to obtain the approximate eigenvalue ...
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1 vote
2 answers
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If set of matrices {AE, BE,CE,DE} linear independent then E is invertible

I'd to to get your help. Let A,B,C,D be matrices in $M_{2}(R)$, prove or disprove the following: If {AE,BE,CE,DE} are different and linear independent then matrix E is invertible: This is my try: Let ...
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1 answer
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A matrix with inner product that respects norm is a reflection matrix

Let $\langle \cdot | \cdot\rangle$ be the standard inner product over $\mathbb{R}^{2}$ and let $A \in M_{2}(\mathbb{R})$ be a matrix such that $\|Av\| = \|v\|$ for every $v \in \mathbb{R^{2}}$ and ...
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0 answers
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A question on a linear injective operator

Let the standard orthonormal basis be given in the $n$-dimensional Euclidean space $$ e_1=(1,0,\ldots,0), $$ $$ e_2=(0,1,\ldots,0), $$ $$ \ldots\ldots\ldots\ldots\ldots\ldots, $$ $$ e_n=(0,0,\ldots,1) ...
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Selection of a regular function in range of matrix

Let $A$ be a $m \times n$ matrix and $f \in C^k([0,1], \mathbb{R}^m)$ ($k \in \mathbb{N}$) such that $f(x) \in \mathrm{Range} \, A$ for almost every $x \in [0,1]$. Can we find a function $g \in C^k([0,...
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1 answer
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Prove that the determinant of the Gram matrix of 3 vectors is nonnegative

Let $\langle\cdot,\cdot\rangle$ to be the regular inner product on $\mathbb{R}^n$ and let $y_1,y_2,y_3\in \mathbb{R}^n$. Prove that $\det(G)\ge0$ where $$ G := \begin{pmatrix} \langle y_1,y_1\...
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0 answers
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Recovering matrix from rotated version

I'm dealing with matrices that came from a software, 3ds Max. It uses 4x3 matrices to represent transformations https://documentation.help/3DS-Max/idx_AT_matrix_representations_of_3d.htm $$\begin{...
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2 votes
5 answers
72 views

Let A be a 2x2 matrix. Prove/disprove if $A^2$=A then either A=0 or A=I

I'm really struggling with this problem. I feel that the statement is true because I can't seem to come up with a specific counterexample. Nevertheless, I don't know how to come up with a proof. Any ...
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-2 votes
2 answers
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Finding a spanning set of a null space [closed]

$$ A= \begin{bmatrix} -3& 6 &-1& 1 &-7\\ 1 &-2& 2& 3&-1\\ 2&-4& 5& 8& -4 \end{bmatrix} $$ Please I have a problem finding the spanning set of a null ...
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2 votes
0 answers
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A similarity-like transfromation with left-invertible matrix.

Suppose $X\in \mathbb R^{m\times n}$ $(m>n)$ is a left-invertible matrix, and its left-inverse is $X^+=(X^\top X)^{-1}X^\top$ (so that $X^+X=I$). Now I have a matrix $A\in \mathbb R^{n\times n}$. ...
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Solution to multidimensional gaussian integral with power factor

I would like to figure out for the derivation for a tensor ABCD optical law was derived for a flattened Gaussian beam in the following reference: https://www.sciencedirect.com/science/article/pii/...
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How can one justify that $\|(I - \mu\mu^T)x\|_2 = \|(I - \mu\mu^T)\|_2\|x\|_2$ where $\mu$ is from the tangent-normal decomposition of the vector $x$?

Full-details about the context of the tangent-normal decomposition are present in this blogpost (search for tangent-normal decomposition). Let $x, \mu$ be $p$-dimensional vectors on the unit sphere. ...
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5 votes
2 answers
84 views

Eigenvalue and spectral condition

Let $A=\begin{pmatrix} 1& 1 \\ a^2 &1 \end{pmatrix} \text{ with } a\in (0,\frac{1}{2}]$. Show $$cond_2(A)=||A||_2 \cdot ||A^{-1}||_2\leq 4(1-a^2)^{-1}$$ by first showing $||A||^2_2\leq||A||_1||...
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1 vote
2 answers
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Trying to understand why a certain matrix derivative is sparse

I'm having a hard time understanding matrice derivatives with respect to derivatives, and came upon the following exercise which I am not sure how to solve. Let there be matrices ${\bf X} \in \Bbb R^{...
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5 answers
126 views

Given matrix $A \in \Bbb R^{n \times n}$ such that $A^2=-I$, find $\det(A)$

Given an $n \times n$ matrix $A$ with real entries such that $A^2=-I$, find the $\det (A)$. My Attempt: $|A|^2=(-1)^n\implies|A|=(-1)^{\frac n2}$ The answer given is $|A|=1$ Eigenvalues are not ...
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2 votes
2 answers
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Over a finite field, which square matrices produce a zero quadratic form?

For which matrices $A \in (\mathbb{F}_p)^{n \times n}$ do we have $x^T A x=0$ for all $x \in (\mathbb{F}_p)^n$? Obviously, this is the case if $A=B-B^T$ for some $B$ (which is equivalent to saying ...
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1 vote
1 answer
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Euclidean projection on convex set of positive semidefinite matrices

Define the Euclidean projection for a convex set $C$ as follows $$\pi_C(y) := \min_{x \in C} \| y - x \|_2^2$$ How would we find the projection map when $C$ is the cone of positive semidefinite ...
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0 votes
0 answers
17 views

Convolution behaviour when applied to matrix product

Consider the discrete convolution operator, applied for example in convolutional neural networks. Let $W\in\mathbb{R}^{k\times k}$ be the convolutional filter and $X\in\mathbb{R}^{m\times m}$ be a ...
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6 votes
1 answer
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Why is the trace of a matrix important?

Lower division linear algebra course at my university taught the simple computation steps to finding a trace of a matrix, but not the intuition nor the purpose for it. What information is gained from ...
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  • 73
0 votes
1 answer
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Proof by induction on matrix [closed]

Can you show, please, how proof this type of exercises by induction. Trying to understand the pattern $$ \begin{pmatrix} 1 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 2 \\ \end{pmatrix}^n=...
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4 votes
1 answer
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Let $A = \bigl( \begin{smallmatrix}2 & 2 \\ -1 & 5\end{smallmatrix}\bigr)$. Prove for $n \geq 1$ using induction that $A^n =$ ...

Can someone check to see if my proof is correct? Feel free to nitpick, trying to get better at writing proofs. Here's the problem: Let $A = \bigl( \begin{smallmatrix}2 & 2 \\ -1 & 5\end{...
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1 vote
0 answers
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How to show the norm of the residual in GMRES minimization problem?

I couldn't quite catch the relationship between QR and it. What I'm confused about is what S and C mean, and how can I make desired proof? Let $e_i^{n} $be the i-th unit vector in $C^n$ The solution ...
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  • 11
-1 votes
1 answer
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Block matrices in MATLAB. [closed]

I am completely new to the matlab programming. Could please tell me algorithm/implementation of the block matrix M(PICTURE). How to code it in Matlab? Thank you.[This Block Matrix M ][1] Block Matrix ...
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6 votes
3 answers
74 views

Remarquable identities $f(n) = \frac{a^n}{(a-b)(a-c)} + \frac{b^n}{(b-a)(b-c)} + \frac{c^n}{(c-a)(c-b)}$

Let $n$ be an integer, and \begin{equation} f(n) = \frac{a^n}{(a-b)(a-c)} + \frac{b^n}{(b-a)(b-c)} + \frac{c^n}{(c-a)(c-b)} \end{equation} \begin{equation} g(n) = \frac{(bc)^n}{(a-b)(a-c)} + \frac{(ac)...
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0 votes
1 answer
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What conditions does positive semidefiniteness impose on the matrix elements?

Let $A$ be an Hermitian $2\times 2$ complex matrix. We can always write it as $$A =\begin{pmatrix}a & \alpha \\ \bar\alpha & b\end{pmatrix}$$ for some $a,b\in\mathbb{R}$ and $\alpha\in\mathbb{...
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  • 5,503
5 votes
2 answers
181 views

Derivative of the inverse of a symmetric matrix w.r.t itself

I'm trying take the derivative of a symmetrix matrix $\mathbf{C}$ with respect to itself. $$ \begin{equation} \frac{\partial \mathbf{C}^{-1}}{\partial \mathbf{C}} \end{equation} $$ Using the indicial ...
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0 votes
1 answer
27 views

Nilpotent map and upper triangular matrix

If we have a map $\phi:V\rightarrow V$ on a vector space $V$ that is nilpotent, then there exists a basis $\underline{\mathbf{v}}$ such that the matrix of $\phi$ with respect to basis $\underline{\...
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0 answers
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Prove for $\|\left[\begin{array}{cc} A&0\\ 0&B \end{array} \right]\|\leq \max\{\|A\|,\|B\|\}$

For this lemma Lemma Let $A$ and $B$ be positive operators on $H$. If $T$ is the operator matrix on $ H ⊕ H$ defined by $T =\left[\begin{align} A&C^*\\ C&B \end{align} \right]$ Then $$\|T\|\...
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-1 votes
0 answers
28 views

Where can I find matrix topology problems?

I am an NBHM aspirant. I am currently studying MSc in Mathematics. In NBHM, I have seen questions from topology of the space of matrices for example compactness, connectedness, openness, closedness ...
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0 votes
0 answers
23 views

Efficiency of seemingly uncorrelated regression estimator against OLS [closed]

The essence is to prove the matrix inequality: $(E[X'\Sigma^{-1}X])^{-1}\le (E[X'X])^{-1}E[X'\Sigma X](E[X'X])^{-1}$ where $E[x]$ is the expectation notation, $\Sigma$ is the covariance matrix and $X$ ...
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1 vote
0 answers
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When does a nonlinear bijective map send independent vectors to independent vectors

Suppose $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, or say a diffeomorphism for convenience. If $\varphi$ is a linear map, it clearly maps linearly independent vectors to linear independent ...
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4 votes
4 answers
116 views

Eigenvectors & eigenvalues of "nearby" matrices [duplicate]

Suppose that I have a square matrix $A$ and another square matrix $B$ whose entries differ by $\varepsilon>0$. Is there any way to bound the differences in their eigenvalues and unit eigenvectors? ...
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1 vote
1 answer
35 views

Finding $2$-dimensional invariant subspace

I just want to check that my understanding is correct of invariant subspaces. I was given a matrix A in which I have found that it is invertible, so I know that a $2$-dimensional invariant subspace ...
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1 vote
0 answers
31 views

Show that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric

In a linear algebra textbook, I was given the following problem: If $B$ is a $n \times n$ square matrix, show that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric. I know that there are relatively ...
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0 votes
1 answer
16 views

Reverse engineering a matrix in RREF

I know that multiple matrices can have the same RREF, but a matrix has a unique RREF, I am trying to reverse engineer a 3x3 RREF matrix to get an abstract matrix of the same size that has 2 columns ...
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0 votes
0 answers
11 views

Prove that a linear partial differential equation of first order can't be elliptic

A linear PDE of first order has the general form $$a(x,y)u_x+b(x,y)u_y-c(x,y)u-d(x,y)=0$$ A PDE is elliptic if the symmetric matrix of the coefficients of the highest derivatives has a determinant ...
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2 votes
0 answers
27 views

Show that $\lambda_j(Q'_nA_nQ_n)-\lambda_j(A_n)\to 0 \quad \text{as}\quad n\to \infty$

Let $(A_n),(Q_n)$ be sequences of $k\times k$ real matrices. Moreover suppose that $(A_n)$ is symmetric and bounded, and that $Q_nQ'_n\to I_k$ as $n\to \infty$, where convergence is with respect to ...
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0 votes
0 answers
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Any thought on that rank similar object?

Context : This problem arises on some exploration around Wyner's common information in information theory and the related minimization problem. Problem : Let $A$ be a $m\times n$ real matrix. Its ...
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1 vote
1 answer
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Derivative of Matrix Power with resepect to entries

Let's consider a matrix $A = \mathbb R^{d\times d}$. I'm interested in the entry wise derivative of $A^n$ that is if $$B = A^n$$ I'd like to find $$ c_{ij} := \frac{\partial}{\partial a_{ij}} b_{ij}.$$...
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1 vote
0 answers
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Difference between basis of a NULL space and a solution space?

I've been tasked to find the basis of the solution space to the matrix $A=\pmatrix{1&-3&1\\2&-6&2\\3&-9&3}$ I've row reduced this matrix to: $\pmatrix{1&-3&1\\0&0&...
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  • 123
-1 votes
0 answers
41 views

Linear algebra Matrix 2x2 [closed]

Let $$ A= \left[\begin{matrix}a&c\\b&d\\\end{matrix}\right] $$ $A$ belongs to $M_{2\times2}(\mathbb{R})$ How to check if it is a diagonalizable matrix and what are the eigenvalues and ...
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