# Questions tagged [matrix-equations]

This tag is for questions related to equations, with matrices as coefficients and unknowns. A matrix equation is an equation in which a variable stands for a matrix .

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### How to write state space equation $\dot{x} = Ax + Bu$ when additive terms are present? [closed]

Suppose there is a system of state $x$ with dynamics: $\dot{x} = Ax +Bu + C$ Where $A$ and $B$ are $n\times n$ matrices showing the dynamics of state $x$ and input $u$. $C$ is a constant $n\times 1$ ...
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### How to relate spectral radius and Cauchy interlace theorem in a separate equation

Consider a a matrix $A_{n\times n}$. Also consider another matrix $M_{(n-1)\times(n-1)}$ which is obtained by deleting the $i^{th}$ row and column of $A$. I have the following equation in $M$ \begin{...
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### derivative of determinant, solve singularity equation for a variable

I would appreciate any tipps on the following problem I really struggle with: For $A \in \mathbb{R}^{n_2\times n_1}, B \in \mathbb{R}^{n_3\times n_2} , \lambda \geq 0, L \geq 0.$ I want to find a ...
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I'm trying to understand the CARE/DARE solution and have a few questions. For example, for the CARE the equation is $$A^tP + PA - PBR^{-1}B^tP+Q=0$$ where $P,Q,R,A,B$ are $n\times n$ matrices and $P,Q,... • 5,279 3 votes 1 answer 50 views ### Equality of product of matrices I have been wondering if the following statement is corrrect? Assume that$A$is$n \times m$matrix where$m \leq n$and rank of$A$is$m$,$X$and$B$are$m \times m$square matrix and it is ... 1 vote 1 answer 44 views ### Computing ABx given Ax In general matrix multiplication does not commute, but is there any information about one ordering of the product contained in a different ordering (assuming the dimensions match)? For example, if I ... • 435 1 vote 1 answer 33 views ### Existence of Solution of a matrix equation Suppose,$Ax=b$does not admit any solution for some$n*n$matrix & some b in$R^n$.Does it imply$A^tx=b$also not admit solution ? I think it may admit solution but I am unable to find out ... 2 votes 1 answer 478 views ### How to solve for$\boldsymbol{\theta}$in$[A]^{\theta}x = b$if$\boldsymbol{[A]}$,$\boldsymbol{x}$, and$\boldsymbol{b}$are known? I am trying to find$\boldsymbol{\theta}$when the output$\boldsymbol{b}$, input$\boldsymbol{x}$, and the matrix$\boldsymbol{[A]}$are given. Here,$\boldsymbol{[A]}$is a diagonal square matrix ... 1 vote 0 answers 29 views ### Is there a such thing as "Stacked Vector Notation"? I was reading this link over here: https://peterroelants.github.io/posts/gaussian-process-tutorial/ . I came across the following statements: I am trying to understand how to "fill in the ... • 1,688 0 votes 1 answer 46 views ### Relationship between a,b and c that make the equation system have infinite solutions i have this matrix \begin{cases} \phantom{2}x_1+2x_2-3x_3=a\\ 2x_1+3x_2+3x_3=b\\ 5x_1+9x_2-6x_3=c\\ \end{cases} And the excercise says that i have to find the relationship between a, b, c that makes ... 0 votes 0 answers 8 views ### Solving tensor version of homogeneous Sylvester equation I have a system that reads (in summation convention) $$X_{j_1 j_2 ... j_N} R^{j_1}_{k_1} R^{j_2}_{k_2} ... R^{j_N}_{k_N} = X_{k_1 k_2 ... k_N}$$ where$N$is a fixed value,$R$a known matrix (... 0 votes 1 answer 21 views ### Proving that matrix to the power multiplied by a non-zero vector are independent from same matrices to different powers Suppose I have a$n$x$n$square matrix$B$and$n$-length column vector$a \ne \underline{0}, such that \begin{align*} B^{3}a &= \underline{0} \\ B^{2}a &\ne \underline{0} \end{align*} I know ... 0 votes 1 answer 39 views ### Weighted least squares formula I was studying the weighted least squares algorithm and came across this formula for calculating the weighted result in terms of the original. Herex_{LS}$is the solution for$D=I$I can't figure ... • 21 0 votes 2 answers 57 views ### If$A$is a square matrix If$A$is a square matrix, and$A=A^2$, then what would the possible values of$|A|$? I've tried to calculate it through basic mathematics, however I feel it's not appropriate... $$A=A^2$$ $$A-A^2=0$$ ... 1 vote 0 answers 19 views ### Finding out one combinations such that linear combinations makes zero vector. Find a combination$~x_{1}\mathbf{\omega}_{1}+x_2\mathbf{\omega}_{2}+x_{3}\mathbf{\omega}_{3}~$that gives the zero vector with$~x_1=1:~$$$\mathbf{\omega}_{1}= \begin{bmatrix} 1\\2\\3 \end{bmatrix}~~~~ ... 0 votes 0 answers 42 views ### How could I calculate the derivate an expression with a diagonal inverse function? Given a vector x=[x_1,x_2,...,x_n] and a matrix Z with dimensions n\times n, the function g(x) is described by:\def\diag{\operatorname{diag}}$$ g(x)=\diag(x) \diag^{-1}(Zx)$$Where \diag(x)... 1 vote 3 answers 141 views ### Locate a pinhole camera using a fiducial marker Note: The superscript notation used refers to the frame of reference. There are three frames of reference: w, the world frame (in Euclidean 2-space), c, the camera frame (in Euclidean 2-space), ... • 19 0 votes 0 answers 25 views ### Derivative of this matrix expression? Assume that I have the following expression:$$f(x,c) = \int_{0}^{x}g(\mathbf{\Psi}(x)^\intercal\mathbf{c}) dx$$where f(x,c) is a scalar, x is a scalar, \mathbf{\Psi}(x) is a N-by-1 vector, ... • 855 0 votes 0 answers 18 views ### Nonsingularity of sum of matrices Consider the matrix E where E is defined as follows: E=A+\gamma BC+DB-BC^{-1}DC where \gamma>0, A,C are positive definite, D is Hurwitz and B is positive semidefinite. I have a ... 0 votes 0 answers 25 views ### Can we give an upper bound for the second largest eigenvalue where there is only one positive eigenvalue? In continuation to my previous post on the nonnegative matrix I have the following follow up question: Suppose A be a strictly positive matrix(entrywise) of the form D+B, where D is a diagonal ... • 11.9k 3 votes 1 answer 87 views ### Which matrix with positive entries has only one positive eigenvalue? Let A be a strictly positive matrix, by strictly positive matrix I mean that the entries of A are strictly positive. Also, we assume that the entries of the matrix are from Natural numbers. Is ... • 11.9k 0 votes 3 answers 87 views ### Are there any solutions to Ax=b satisfying \Vert x \rVert=1? [closed] Let$$A = \begin{pmatrix} 1 & 1 & -3 \\ -3 & -2 & -2 \\ -7 & -5 & 1 \end{pmatrix}$$be a 3 by 3 matrix and$$b = \begin{pmatrix} 4 \\ 6 \\ 8 \end{pmatrix}$$be a column ... 0 votes 1 answer 29 views ### insert a matrix in-between product of vectors Let a=[1,2]^t ( ^t means transpose) and let A=\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}. One can verify that a^t a = 5, a^t A a = 14, and a^t a \cdot \det(A)= 15. So obviously, ... • 23 4 votes 1 answer 44 views ### When does the following frobenius norm equality hold? When we consider the square matrix {\bf A}\in\mathbb{C}^{N\times N}, then following ineqaulity always holds:$$ \left\|{\bf A^{\sf H}}{\bf A}\right\|_F = \left\|{\bf A}{\bf A^{\sf H}}\right\|_F\le\... • 43 0 votes 0 answers 21 views ### Having trouble understanding the notation for matrices with seemingly incompatible dimensions in a paper; dimensions for matrix math? Looking at equation 4 in this paper. $$H_{aj}^{k+1} = H_{aj}^{k} \frac{\sum_{i=1}^{n}W_{ia}^{k}V_{ij}/(W^{k}H^{k})_{ij}}{\sum_{i=1}^{n}W_{ia}^{k}}$$ For more context, this is a non-negative matrix ... • 1 10 votes 1 answer 145 views ### If square matrices$A^2 + B^2 = 2AB$, then prove that$p_A(x) = p_B(x)$Original problem statement: Let$A, B \in M_n(\mathbb{C})$such that$A^2 + B^2 = 2AB$. Prove that for any$x \in \mathbb{C}$: $$det(A - xI_n) = det(B-xI_n)$$ Now the first observation, the equality ... • 871 0 votes 0 answers 28 views ### Formula for matrix inverse with non-commutative entries [duplicate] I have a square matrix$A$with elements$A_{i,j}\in\mathbb{A}$where$\mathbb{A}$is a ring with with addition ($+$) and multiplication operations ($\times$). The operation$\times$is non-... • 563 0 votes 1 answer 43 views ### How to solve the matrix equation$\mbox{tr}(\mathbf{X}\mathbf{X}^H)\overline{\mathbf{B}}=\mbox{tr}(\mathbf{X}\mathbf{X}^H\mathbf{B})\mathbf{I}$? I want to solve the following equation for$\mathbf{X}\in\mathbb{C}^{N\times M}$, with$M < N$: $$\mbox{tr}(\mathbf{X}\mathbf{X}^H)\overline{\mathbf{B}}=\mbox{tr}(\mathbf{X}\mathbf{X}^H\mathbf{B})\... • 41 0 votes 1 answer 21 views ### 2 identities related to determinant and exponential function This question is from my assignment in smooth manifolds and I was unable to solve this problem. I am asking it here as I think I will not be able to solve it by myself. Question : Let A\in M(n, \... • 1,497 1 vote 1 answer 14 views ### Equivalent parametric solution sets I have this question that I've never seen before, I've only ever learned about how to find the parametric version of a solution set, but I've never learned how to change it in any way... I will type ... 0 votes 1 answer 46 views ### Minimal polynomial of the matrix A = \begin{bmatrix} c & 1 & 0 & 0\\ 0 & c & 0 & 0 \\ 0 & 0 & c & 1\\ 0 & 0 & 0 & c \end{bmatrix} I am learning about the minimal polynomial of a matrix for the first time, but I don't understand how to find quickly the minimal polynomial of some matrices. For instance, I know that the minimal ... • 521 0 votes 0 answers 19 views ### Calculating the unknown elements of a matrix based on a known matrix. Suppose there is a 2\times2 matrix M=\matrix[a\ ,b\ ;\ c\ ,d], which$$M \matrix[y_1\ ,\ y_2]^T=\matrix[y_1'\ ,\ y_2']^T$$if the elements of the matrix M are known, is it possible to calculate ... • 23 1 vote 1 answer 31 views ### How to decompose matrix to its composition of rotation and scaling? Ans: for rotation and scaling factor are$\frac{\pi}{6}$and 8 respectively. I found a related question, but it wasn't explaining it well. I also understand scale and rotation separately but cannot ... • 643 0 votes 1 answer 22 views ### General term for a series of matrix multiplication A and B are two non-zero square matrices such that${A^2}B=BA$and if$(AB)^{10} = {A^k}{B^{10}}$then the value of k is? Attempt: Tried solving for lower powers of$AB$and observing a pattern$AB=AB$... 0 votes 1 answer 32 views ### Calculator gives unit vector, how to derive vector given only unit vectors? My CAS HP Prime calculator gives only unit vectors when I use the function for Eigenvectors. I know how to find a unit vector,$u=\frac{v}{||v||}$, but that doesn't seem to help. For example: ... • 643 0 votes 1 answer 57 views ### Diagonalization of$A=\begin{pmatrix}8&12&-18\\ 4&18&-20\\ 4&13&-15\end{pmatrix}$such that$A=PDP^{-1}$, how to find P, specifically,$P_2, P_3$Using wolfram and symbolab, diagonalization of$\begin{pmatrix}8&12&-18\\ 4&18&-20\\ 4&13&-15\end{pmatrix} \rightarrow A=PDP^{-1}\rightarrow \begin{pmatrix}8&12&-18\\ 4&...
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a. I got the change of coordinates from B to A, P = $\begin{pmatrix}6&0&-6\\ -1&4&0\\ 1&1&3\end{pmatrix}$ b. However, I tried to use the formula for change of basis \$P^{-1}(x)=(...