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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How to show the derivative of $f(M)=\text{Tr}(M\log (M) -M)$ is $\log (M)$?

Let $M$ be a positive definite matrix in $\mathbb{S}_+^n$. Let $\log$ be natural matrix logarithm which $\log(M)$ is defined as $\log(M)=\sum_{i=1}^{n}\log(\lambda_i)v_iv_i^T$ where $(\lambda_i,v_i)$ ...
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Suppose $p(x)=ax^2+bx+c$ is a polynomial where $a, b$, and $c$ are fixed but unknown constants. Setup and solve a $3 \times 3$ system of linear equations to find $a,b$, and $c$ if we know that $p(1)=4,... 0answers 30 views Borel subgroup of$SL_2(\mathbb{Z})$As the title indicates, I want to ask what is the Borel subgroup of$SL_2(\mathbb{Z})$? I believe I read about it in one of James Milne's notes. But now I cannot find it. That is why I want to ask. 0answers 5 views How to avoid scaling issues when rotating a child matrix, if its parent matrix has a nonuniform scale? If a parent transform has a nonuniform scale, then I try to rotate its child transform locally, the child is sheared weirdly. How can I avoid that? 1answer 37 views Why taking integral of both sides of matrix inequality is allowed? How to show if$\nabla^2 f(x) \succeq \alpha I$, then the function is$\alpha-strongly convex? In my optimization notes I have \nabla^2 f(x) \succeq \alpha I \rightarrow \alpha\text{-strongly ... 4answers 87 views How to prove that (AB)^t = B^tA^t The proof given in my book (and I came up with as well) is: However, the part that throws me off is line #3 where they do \Sigma A_{jk} B_{ki} = \Sigma B_{ki} A_{jk} I understand that ... 1answer 24 views Question about matrix equation and it's simplification For example, we had such an equation (where A is a matrix, and we need to find X, also where T is transpose) - A = (2 \cdot X)^T So, we will have something like this? According to the sources ... 0answers 9 views Converting a transformation from world frame of reference to local? I have a matrix W that is the local-to-world transformation (Rotation/Scale/Position) of a point P in 3D space. I also have a rotation quaternion R and a translation vector T that further ... 1answer 43 views Some questions about simplification of the equation with matrices I have such an equation (X^{-1} \cdot 2 \cdot A)^{-1} - B = 4X Where B and A are matrices. So, I did such a simplification. \begin{align} & \frac 12 A^{-1}X - B = 4X\\ \iff & \frac 12 ... 2answers 51 views Diagonalizing matrix with fractions [on hold] I'm revising for an exam in linear algebra, and I've found myself stuck on this one specific exercise. I'm supposed to decide a matrix P and a diagonal matrix D from my matrix H (which I'll ... 1answer 28 views Null space of a rotation matrix If we have a rotation matrix of the kind: link to the rotation matrix how do i compute the null space of this matrix? I know that to obtain the null space we need to write the matrix in echelon ... 1answer 31 views Prove that for \pi_g(\alpha)=\alpha g, \pi is a group action Let R a principal ideal domain and let G the group of all the invertible members of M_2(R). Let \Omega :=R^2. For all g\in G let \pi_g:\Omega\to\Omega defined by \alpha \pi_g=\alpha g ... 0answers 58 views How do I invert this matrix? Given two vectors\vec{v} = \begin{pmatrix} v_1\\ \vdots\\ v_n \end{pmatrix} , \vec{w} = \begin{pmatrix} w_1\\ \vdots\\ w_n \end{pmatrix} \in \mathbb{R}^n$$such that ... 1answer 22 views Matrix units in commutator subgroup Let m \ge 2 be an integer, R = M_{m}(K) be the algebra of all m\times m matrices over K an infinite field, if L = [R,R] and a,p \in R are non-zero elements. Then, if [au, pu] = 0 for all ... 0answers 20 views Conjugacy classes in special linear groups over \mathbb{Z}/p^k Let n be an integer and p^k a prime power. Question: Is the number of conjugacy classes of elements in \mathrm{SL}_n(\mathbb{Z}/p^k) known? Remark: An answer for n=2 would already help me. 0answers 13 views Expression for maximum column index of every row in matrix Let H is (k \times j) matrix. How can one describe "for every maximum column of each row in H"? Can I write \underset{j}{\text{argmax}} (H_{kj}), {k = 0,\ldots,K}? I feel something is wrong ... 1answer 27 views Augmented matrix in equation form I recently learned about using an augmented matrix to find the inverse of a matrix by putting the entire identity matrix on the right side of an augmented matrix and getting the left side to reduced ... 0answers 12 views Why does finding the partial derivative solve this payoff matrix? So I'm taking this math course and I decided to do all the homework the first day of class, and as I was pulling an all-nighter doing it, I got to the topic of game theory and payoff matrices. I never ... 0answers 7 views Applying a Sigma-Point Kalman Filter to State of Charge Estimation I've found a research paper that would allow me to implement a Kalman filter to estimate state of charge for a LiPo battery, but cannot make sense of some of the symbols as I'm a first year. Can ... 1answer 15 views Computational complexity of finding determinants What is the computational complexity of finding the determinant of a matrix in this form? \begin{bmatrix} x_{1,1} & x_{1,2} & x_{1,3} & \dots & x_{1,n-1} & x_{1,n} \\ x_{... 1answer 32 views Let A be 2 \times 2 nonzero real matrix.which of the following is true? Let A be 2 \times 2 nonzero real matrix.which of the following is true? (A) trace of A^2 is positive (B) A has non zero eigenvalue. (C) All entries of A^2 can't be ... 1answer 36 views Is there any inequality between 2-norm condition number and Frobenius norm condition number for rectangular matrix? What I have found in [1] Condition number inequality between Frobenius norm and 2-norm for square matrix, Consider a full rank matrix X \in \mathbb{C}^{n \times m}, m=n, then we can have,$$n - ... 1answer 69 views Fast way to Invert ADA' when D is a diagonal matrix that changes each iteration? So I have a statistical learning algorithm in which D is a diagonal matrix that changes each iteration while A stays the same. I'm looking for a fast way to invert ADA' each iteration which ends up ... 0answers 23 views Gauss-Seidel: I can't get diagonally dominant equations I've got the question: Use the Gauss-Seidel method to solve:8x_1 - 16x_2 = 64x_1 + 8x_2 = 0$I know that the equations have to be diagonally dominant, but I can't see how to determine this. I ... 2answers 38 views Can$\| \left( X'X + \lambda I \right) ^{-1} X'y \| = t$be solved for$\lambda$? In this post I suggested that the expression $$\| \left( X'X + \lambda I \right) ^{-1} X'y \| = t$$ couldn't be easily solved for$\lambda$, because you need to "invert" the norm. But in general ... 1answer 25 views Determine the values of a, b for which the systems have (1) exactly one solution, (2) no solutions, (3) infinitely many solutions. I'll leave two pictures, can someone check if I'm right? (exercise b) 0answers 22 views What kind of matrices can be visualized by graphing the column vectors? In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis ... 1answer 20 views Determine the values of a, b and c, for which the systems have (1) exactly one solution, (2) no solutions, (3) infinitely many solutions. I will just attach a picture. Can someone help me to solve this? I think I missed some information. 2answers 19 views Let$v \in R^{k}$,with$v^{T} v \neq 0$. Let$P=I-2 \frac{v v^{T}}{v^{T}v}$, where$I$is the$k X k $identity matrix. Let$v \in R^{k}$,with$v^{T} v \neq 0$. Let$P=I-2 \frac{v v^{T}}{v^{T}v}$, where$I$is the$k \times k $identity matrix. Then prove that eigenvalue of$P$are$1,-1$and$P^{2}=IP^{2}=(I-2 \...
I have some equations and I don't know am I doing the simplification right. Can someone check it? For example, we have an equation $AX = 4X + B$, where $A$ and $B$ are matrices. So, what I have done ...