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
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1answer
198 views

Proving that the sum of the errors of a least square linear approximation is $0$

Let $(x_1,y_1),\dots,(x_n,y_n)$ be points in $\mathbb{R^2}$ and $e=[\epsilon_1,\dots,\epsilon_n]^T$ the error vectors belonging to the least square solution of the linear approximation. Prove that ...
2
votes
2answers
2k views

How to calculate what ratio of matrix rows will, when summed, equal another row?

I'm sorry for my ignorance; I don't even know the correct terminology or proper way to phrase my question. I tried searching but require human assistance. I'm sorely deficient in understanding of ...
2
votes
1answer
178 views

Cholesky decomposition given an initial factorization $RR^{T}$

Given a rectangular matrix $R\in\mathbb{R}^{m\times{}n}$, I am interested in computing the Cholesky factorization of $RR^{T}$. From what I understand the decomposition requires $\frac{1}{3}m^3$ FLOPS ...
2
votes
1answer
123 views

which of the following options is correct?

Let $y(t)=\begin{pmatrix} y_1(t)\\ y_2(t) \end{pmatrix}$ satisfy $\dfrac {dy}{dt}=Ay; t>0, y(0)=\begin{pmatrix} 0\\ 1 \end{pmatrix}$ where $A$ is a $2 \times 2$ constant matrix with real ...
2
votes
1answer
173 views

Jordan basis of $A$ when $A$ is the companion matrix?

The actual question: when $A$ is the companion matrix, why the general form of $M_i$ (the group of columns of the Jordan matrix $M$ that belongs to the block associated to $\lambda_i$) is: $$ ...
2
votes
1answer
84 views

Question on linear algebra-matrices

Let $A$ be an $m\times n$ matrix and $B$ an $n\times k$ matrix. Show that the columns of $C=AB$ are linear combinations of the columns of $A$. If $\alpha_1,\dotsc,\alpha_n$ are the columns of $A$ and ...
2
votes
1answer
136 views

Rivlin-Ericksen theorem (Order $n$)

[Ciarlet 1.3-10] Denoting by $\mathscr{S}_n(\mathbb{R})$ and $\mathscr{O}_n(\mathbb{R})$ the sets of symmetric and orthogonal matrices respectively, of order $n$, show that a function $$\mathscr{H} : ...
2
votes
1answer
205 views

Topologically equivalence of a metric on matrices

Define a function on the set of $n\times n$ matrices by $\rho(A,B)=\operatorname{rank}(A-B).$ Prove that $\rho$ is a metric that is topologically equivalent to the discrete metric.
2
votes
1answer
87 views

Disable one angle of rotation

I'd like to disable one angle of rotation of an object rotating in 3D space. Imagine a camera rotating around and displaying objects as they are in space. I'd like this object to be fixed on the ...
2
votes
1answer
290 views

How can I convert a NxN Matrix to a Vector Nx1?

$$ \left[ \begin{array}{@{}ccccc@{}} 0.9& 0.1& 0& 0& 0& 0& \\ 0& 0.9& 0.1& 0& 0& 0& \\ 0& 0& 0.9& 0& 0& 0.1& \\ 0& 0& ...
2
votes
2answers
131 views

How to show for a PSD matrix $A$ that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$?

If $A \in \mathbb{C}^{n \times n}$ is positive semidefinite, show that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$, where $\sigma _{\min}\left ( A ...
2
votes
3answers
37 views

Matrix multiplication distributivity

Suppose we have matrices $A, B, C$ of dimensions $m \times n, m\times n, n \times l$ respectively. How can we prove $(A+B)\circ C = A\circ C + B \circ C$ (using the summation notation method?)
2
votes
1answer
58 views

If $Tr(A^k)=0$ for $1\leq k\leq n$ is $A^n=0_n$, where $A\in M_n(\mathbb R)$?

I know that the relation $Tr(A^k)=0$ is equivalent to $\sum_1^n(\alpha_i^k)=0$ where $\alpha_i$ is an eigenvalue of A and that $\alpha_i=0$ for any i is a solution to the system, but is it unique? ...
2
votes
1answer
62 views

How can I solve a polynominal of degree 2 with more than one variable?

(Sorry if the title is not informative) How can I find the value of matrices $F$ and $d$, in the following equation: $$y'Ay+b'y+c'c = (y-d)'F(y-d)$$ Given $A:n \times n$, which is positive definite ...
2
votes
1answer
155 views

Transforming matrix-equation to overdetermined minimum problem

i have broken down my problem to plainmath and could really use some help. Basis: I have an image. In this image I have several UV-XYZ pairs. So i know the 3d position of serveral Pixels. Given the ...
2
votes
1answer
187 views

Regarding the kernel of a linear transformation and that of the associated representing matrix

Let $V, W$ be finite dimensional vector spaces over a field $F$. Let $\mathcal{B}_{V} = \{\mathbf{v_1, \cdots, v_n} \}$ and $\mathcal{B}_{W} = \{\mathbf{w_1, \cdots, w_m} \}$ be corresponding bases. ...
2
votes
1answer
35 views

Bounding the dimension of a subspace associated with a hermitian form.

Suppose that $H$ is an invertible hermitian (self adjoint) matrix in $\mathbb{C}^{n×n}$. Define the hermitian (sesquilinear) form $[x,y]:=x^∗Hy$. Suppose $V$ is a subspace of $\mathbb{C}^{n}$ ...
2
votes
1answer
285 views

Find the normal vector to the projection plane

In the context of perspective projection. Given focal length is $2.387$, the camera is at $(0.0.0)$ looking at $-z$ direction A rectangle lies on a plane tilted from view plane. Also given the ...
2
votes
1answer
205 views

How to compute the projection $P(v)$ of a vector to a subspace? How to find the matrix of $P$?

Let $P(v): \mathbb{R^4} \to\mathbb{R^4}$ be the projection of a vector $v$ to the space $\left\langle\left|\begin{smallmatrix} 1 \\ 2\\ 2\\ 1 \end{smallmatrix}\right|, \left|\begin{smallmatrix} ...
2
votes
1answer
118 views

Trace matrix inequality

Let $A,B$ be positive definite matrices, and assume that $$ a_{i,j}<{b_{i,j}} $$ for all $1\leq i,j\leq n$, where $a_{i,j}$ is the $(i,j)$ element of the matrix $A$ and $b_{i,j}$ is the $(i,j)$ ...
2
votes
1answer
81 views

For what A, If $A+A^T>0$ then $A^2+A^{2T}>0$?

let me know if I am wrong with the next with a real square matrix A. $A+A^T = \sqrt{A^2+A^{2T}+AA^T+A^TA} > 0$ This square root exists right? And because of this, the sum of all its elements are ...
2
votes
2answers
109 views

How do I know if a discrete time-invariant homogeneous dynamic system will reach, at some point, an equilibrium point?

Is this even possible? Given a time-invariant homogeneous dynamic system: $$x(k+1) = Ax(k)$$ My textbook defines an equilibrium point of the system as: A vector $\bar x$ is an equilibrium point ...
2
votes
1answer
149 views

Derivative of the off-diagonal $L_1$ matrix norm

We define the off-diagonal $L_1$ norm of a matrix as follows: for any $A\in \mathcal{M}_{n,n}$, $$\|A\|_1^{\text{off}} = \sum_{i\ne j}|a_{ij}|.$$ So what is $$\frac{\partial ...
2
votes
1answer
4k views

How to find a transformation matrix having several original points and their respective transformed results?

I have three original points $pt_1, pt_2, pt_3$ which if transformed by an unknown matrix $M$ turn into points $gd_1, gd_2, gd_3$ respectively. How can I find the matrix $M$ (all points are in ...
2
votes
1answer
79 views

Is there a matrix satisfying a certain condition

I have a numerical problem which boils down to the following: We are given a square matrix $R$, with a bunch of zeros in it. We want to check if there exists orthonormal matrix $T$ such that $TT'=I$, ...
2
votes
2answers
58 views

assume $A\in M_n(\mathbb C)$,$A^2=0$ how prove $\exists C,B\in M_n(\mathbb C)$ such that A=BC and CB=0?

assume $A\in M_n(\mathbb C)$,$A^2=0$ how prove $\exists C,B\in M_n(\mathbb C)$ such that $A=BC$ and $CB=0$. Thanks in advance
2
votes
1answer
93 views

Derivative of $F(Ax)$

What is the identity for $$ \frac{\partial \mathbf{F}(\mathbf{A}\mathbf{x})}{\partial \mathbf{x}} = ?$$ If $\mathbf{A} \in \mathbb{R}_{mn}$, $\mathbf{x} \in \mathbb{R}_n$, and $\mathbf{F}: ...
2
votes
3answers
341 views

How to solve a system of 3 equations with Cramer's Rule?

I am given the following system of 3 simultaneous equations: $$ \begin{align*} 4a+c &= 4\\ 19a + b - 3c &= 3\\ 7a + b &= 1\end{align*} $$ How do I solve using Cramers' rule? For one, I ...
2
votes
2answers
1k views

Finding a congruent matrix

I have the matrix $$A =\begin{pmatrix}0&1\\1&0\end{pmatrix}$$ How would I diagonalize it using elementary row operations? It's been a while since I've worked with them so I'm doubting ...
2
votes
1answer
81 views

Assume $A_1,A_2,…,A_n\in M_{m×m}(F)$ that satisfy the following conditions, how to prove that $A_1A_2…A_n=0$?

Assume $A_1,A_2,...,A_n\in M_{m×m}(F)$ (where $F$ is a field) such that $A_jA_i=A_iA_j$ $A_i^2=0, \;\;\forall 1\leq i \leq n.$ If $m\lt2^n$ then how to prove that $A_1A_2...A_n=0.$ Thanks in ...
2
votes
3answers
47 views

Image and vectors question?

So I have the image $\displaystyle \operatorname{Im}f=\{\lambda_1(1, 2 ,0)+\lambda_2(2 ,1, 3) \}$ and I have to find the values of $\lambda$ so that the vector $\displaystyle (1,\lambda,\lambda^{2}) ...
2
votes
1answer
311 views

Relation between trace and Ky Fan norm

As we know that the Ky Fan k Norm is the sum of k-th largest singular values. On the other hand, the trace of a matrix is the sum of its eigenvalues. For a N by N symmetric matrix $M$, its Ky Fan ...
2
votes
3answers
104 views

How can we expand a matrix while maintaining the entries of its inverse?

Let $A$ be an invertible $n \times n$ matrix. Fix some $m > n$. An invertible $m \times m$ matrix $B$ inverts with $A$ if the principal submatrix consisting of the first $n$ rows and cols of ...
2
votes
2answers
345 views

Can Every Square Matrix be written as product of two commuting matrices.

The title explains it all. Can every square matrix $A$ be written as $A=B_1B_2=B_2B_1$ of any two matrices $B_1$,$B_2$.
2
votes
1answer
236 views

Non linear phase portrait

Consider the (nonlinear) system, $ \left\{ \begin{array}{l} \dot x = \left| {y\left| {} \right.} \right.\\ \dot y = - x \end{array} \right.$ Sketch the phase portrait of the system. I have ...
2
votes
1answer
59 views

To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
2
votes
1answer
88 views

Notation of Matrix and Coordination

I was confused by the notation of the following question Let $E = I_5(R_2\leftarrow R_2+4R_3)$, then $E^{-1}=I_5(C_p\leftarrow C_p+\alpha C_q)$, what are the values of $p,q,\alpha$? I know that ...
2
votes
2answers
65 views

Stable method to compute $A^n$ for this defective matrix $A$?

I'm looking for a stable method to compute $A^n$, where $A$ is the following defective $12 \times 12$ matrix: $$A = \left(\begin{array}{cccc|cccc|cccc} \frac{1}{2} & \frac{1}{2} & 0 & 0 ...
2
votes
1answer
62 views

$\int_{\mathbb{R}^n} dx_1 \dots dx_n \exp(−\frac{1}{2}\sum_{i,j=1}^{n}x_iA_{ij}x_j)$?

Let $A$ be a symmetric positive-definite $n\times n$ matrix and $b_i$ be some real numbers How can one evaluate the following integrals? $\int_{\mathbb{R}^n} dx_1 \dots dx_n ...
2
votes
1answer
70 views

$GL(2,\mathbb{R})$ as a subset of $\mathbb{R}^4$

If we consider $GL(2,\mathbb{R})$ as a topological subspace of $\mathbb{R}^4$ with the usual topology and want to know if it compact or not then if we could show that it was not closed then we would ...
2
votes
1answer
1k views

Lower bound on norm of product of two matrices

Let $\vert \vert . \vert \vert$ be the 2-norm. Since this norm is submultiplicative, we know that for any two square matrices $A, B \in \mathbb{R}^{n \times n}$, $$ \vert \vert A B \vert \vert \leq ...
2
votes
1answer
239 views

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
2
votes
2answers
506 views

Rotation matrix in terms of axis of rotation

How to calculate the rotation matrix in 3D in terms of an arbitrary axis of rotation? Given a unit vector $V=V_{x}e_{x}+V_{y}e_{y}+V_{z}e_{z}$ How to calculate the rotation matrix about that axis?
2
votes
1answer
504 views

What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ...
2
votes
2answers
396 views

Smallest EigenValue and Frobenius Norm

The matrices I discuss are all $N\times N$ hermitian matrices. Consider two (hermitian) matrices $A_1$ and $A_2$. For a real scalar $t$, define the following function for the matrix $A_1+t*A_2$ ...
2
votes
1answer
3k views

how to prove a symmetric matrix is positive semidefinite?

I have a symmetric matrix where the diagonals are all positive. I need to prove the matrix is positive semidefinite. The matrix is made up of a bunch of constants and I tried getting the eigenvalues ...
2
votes
1answer
306 views

Geometric multiplicities of the same eigenvalue of $A$ and of $A^T$

For a square complex/real matrix $A$, $A$ and $A^T$ have the same set of eigenvalues, each with same algebraic multiplicities, since their characteristic polynomials are the same. I wonder for each ...
2
votes
1answer
347 views

Need help with relative and absolute errors?

Lets assume I have $Ax=b$ equation, where $A$ is $2$x$2$ matrix. 1) I want to find an A, x, and b such that relative error in x is small but absolute error in x is large 2) Also want to find A, x, ...
2
votes
1answer
80 views

$QR$ decomposition of matrix

Let $Q_1$, $Q_2$ be unitary matrix and $R_1$, $R_2$ be upper triangular with positive diagonal elements. How do I prove that if $Q_1 R_1=Q_2 R_2$, then $Q_1=Q_2$ and $R_1=R_2$?
2
votes
1answer
218 views

Linear Algebra Question ( rank of matrix )

Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively prove $\operatorname{rank}(\mathbf{PA}) = ...