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 .
4,413
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Integrate product of matrix exponentials of a symmetric matrix
Let $\mathbf{A}$ $\in \mathbb{R}^{N \times N}$ be a real, invertible, symmetric matrix.
Let $\mathbf{Q}$ $\in \mathbb{R}^{N \times N}$ be a real, invertible matrix.
Given these properties of $\mathbf{...
- 41
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Comparison of Solution Norms for Invertible and Near-Singular Matrices Using Pseudoinverses
I am studying the behavior of solutions to linear systems of equations where the coefficient matrices are either invertible or near-singular. Specifically, I am interested in understanding the norms ...
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Characterising the nature of Eigen values of a given paramteric matrix
question: A{2x2} = https://files.oaiusercontent.com/file-rZmIDJV1d2325YK6eQfPFTo1?se=2024-07-18T09%3A30%3A45Z&sp=r&sv=2023-11-03&sr=b&rscc=max-age%3D299%2C%20immutable%2C%20private&...
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Problem on Right Inverses of Matrices
Let $n,m$ be integers with $n \leq m$. Let $A, B$ be $n \times m$ real matrices of rank $n$ (i.e., of full rank).
I would like to show that there exists an $m \times n$ matrix $C$ such that:
$AC = ...
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Can one solve this complex linear equation
Suppose the equations are as follows:
$$\text{tr}\{AH\}= c$$
where $c \in \mathbb{C}$ and $A$ is a symmetric matrix in $M_{N}(\mathbb{R})$ are both known and $H$ is a Hermitian matrix which looks like:...
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Does an equivalent of the condition number for the error in system coefficients of a linear equation system exist
Let's assume that a system
$$
[A]\overline{x} = \overline{b}
$$
is given.
The normal condition number assigns an upper bound to the relative error in $\overline{x}$ based on the error in $\overline{b}$...
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Proving that the rank of the following matrix is $6$.
In my research work I have come across a matrix which has the rank equals to $6$. I begin defining my problem as follows: Let $P \in \{0,1\}^{7 \times 7}$ denote the right shift matrix defined by
$ P =...
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Is there a Cauchy-Schwarz inequality for $Tr[A B C D]$?
I am familiar with the Cauchy_Schwartz inequality
$$|Tr[A^*B]|^2 \le |Tr[A^*A]| |Tr[B^*B]|$$
where $*$ denotes the conjugate-transpose operation. I am wondering if there is a similar inequality for $...
- 83
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What mathematical terminology and equations are used for variant assertions of finite sets?
Below there is a set of variants (or enumerations) of multiple finite sets with a different number of items in each set.
I don't want to use the word combinations because I believe those are of the ...
- 462
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The matrix equation of the form $AX+XA^{-T}=0$ has a nonsingular anti-symmetric solution $X$
I want to prove that for $A=J_n(i)$, that is, the Jordan block matrix corresponding to the eigenvalue $i$ of size $n$, where $n$ is even, the matrix equation $AX+XA^{-T}=0$ has a nonsingular anti-...
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Finding All Linear Factors of a Determinant for a Given 3x3 matrix
I am working on finding all the linear factors of the determinant of the following $3\times 3$ matrix:
\begin{vmatrix}
a & a^3 & a^4 \\
b & b^3 & b^4 \\
c & c^3 & c^4
\end{...
- 101
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coordinates of the vector relative to the new basis
Given vectors $(v, b_1, b_2, b_3,...,b_n)$ defined by their coordinates in an arbitrary basis. Prove that the vectors $(b_1, b_2, b_3, \ldots, b_n)$ form a basis and find the coordinates of the vector ...
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Any decomposition of inverse of nonnegative diagonal matrix times a PSD matrix plus lambda times Identity?
I generally have to solve the following system:
$$
(DA + \lambda I)^{-1} v
$$
where $D$ is a diagonal matrix with nonnegative entries, $A$ a symmetric, positive semi-definite (PSD) matrix, $I$ is the ...
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Write the Kalman state vector update equation into the form $Ax=b$
I was reading an old article on Kalman Filters (https://doi.org/10.1109/TAC.1983.1103242),
and they state that the state vector corrector equation:
$$\hat x = \bar x + PH^T (HPH^T+R)^{-1} (z-H\bar x)$$...
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Clarification on the forward extended Euclidean algorithm for finding gcd and linear solution [duplicate]
I have been reviewing Bill Dubuque's explanation of a forward version of the extended Euclidean algorithm in another question. I have seen other explanations of this method on the internet, but Bill's ...
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