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.
57,483 questions
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Let A be an m×n matrix of rank r. Prove that $PAQ=\begin{pmatrix} I_r &0\\0 &0 \end{pmatrix}$ [closed]
Let A be an m×n matrix of rank r. Prove that there exist invertible square matrices P and Q of order m and n, respectively, such that
$PAQ=\begin{pmatrix} I_r &0\\0 &0
\end{pmatrix}$
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5
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2
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When does the inverse for a sequence of matrices exist?
Let $A_n$ be sequence of $n\times n$ invertible matrices over $\mathbb{C}$, and $A_{n+1}(i,j)=A_{n}(i, j) \ \forall 1\le i,j\le n$ (i.e we obtain $A_{n+1}$ from $A_n$ by adding a new row and column at ...
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0
votes
0
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LDL decomposition when D has different size
Typically the LDL decomposition is formulated for L and D being square (lower triangular and diagonal respectively) matrices of the same shape as the symmetric positive-definite input:
$$ L D L^T = A $...
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1
vote
2
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84
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Finding rotations transform
Two vectors vec1 and vec2 were randomly rotated the same way about origin. The result of rotations are vectors ...
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3
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0
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Finding the degree $d$ of the minimal polynomial of $A$ by finding a vector $X$ such that $X,AX,\ldots,A^{d-1}X$ are linearly independent
Let $A$ be an $n \times n$ matrix over a field $F$. Let $d(A)$ be the largest non-negative integer such that the matrices $Id, A, A^2,\ldots,A^{d(A)}$ are linearly independent in the vector space of ...
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2
votes
2
answers
77
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Finding $CB$ given $BC=A(B−A)^{-1}$ for non-singular matrices
I am trying to solve a linear algebra problem from a practice exam.
The Problem:
Consider three square matrices $A$, $B$, and $C$, all of order $n$, with real entries. Assume that each of them is non-...
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1
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0
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Decomposition $PLUP^{-1}$
Let $A\in\mathrm{M}_n(\mathbb{C})$. If the leading principal minors (namely the determinants of the top left submatrices) of $A$ are non-zero, then there exists a lower triangular matrix $L$ and an ...
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votes
1
answer
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On the invertibility of leading principal submatrices in orthogonal matrices with strict column diagonal dominance [closed]
Suppose we have a real unitary (orthogonal) matrix in which the first three diagonal entries are strictly diagonally dominant within their respective columns, meaning each is strictly greater in ...
1
vote
0
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Eigenvalues of real left-circulant matrices
Consider the $3\times3$ left-circulant matrix with first row $(0,1,1)$:
$$ C = \begin{pmatrix}
0 & 1 & 1 \\
1 & 1 & 0 \\
1 & 0 & 1
\end{pmatrix} $$
where each row is a left ...
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2
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Solving linear system with triangular Toeplitz matrix
I'm searching for the solution $\bar{a}$ to the system of equations $\bar{e}_1 = B\bar{a}$ given by
\begin{equation}
\left[\begin{array}
& 1 \\
0 \\
\vdots \\
0 \\
0
\end{...
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2
votes
1
answer
92
views
The opposite category $\mathbf{Mat}_R^\text{op}$
For a commutative ring $K$, the category $\mathbf{Mat}_K$ is defined as follows:
objects are positive integers $m, n, ...$
morphisms $A \colon n \to m$ are $(m \times n)$-matrices, and for two ...
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votes
0
answers
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Finding change of basis matrices in $\mathbb{R}^2$ [closed]
In $\mathbb{R}^2$, consider the following ordered basis:
$$B=((1,0),(0,1)),\text{ }C=((-1,1),(1,1)),\text{ and }D=((\sqrt{3},1),(\sqrt{3},-1))$$
Find the change of basis matrix from $B$ to $C$, from $...
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2
answers
200
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Checking if the determinant of a quasi-tridiagonal matrix is nonzero
From a homework assignment for an undergraduate course on numerical methods:
Check if the determinant of the following matrix is nonzero.
\begin{bmatrix}
1 & -2 & 1 & & & &...
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0
votes
1
answer
86
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This seem to be a sufficient condition for non-commutative matrix product to be symmetric. Is it also necessary? [closed]
I observed that for matrices of following forms, their product are also symmetric:
a b c
b c b
c b a
or perhaps also other similar forms.
Truth to be told, I'm ...
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votes
1
answer
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Connection between two methods of finding the inverse of an invertible matrix
I've learned two methods of calculating the inverse of an $n \times n$ non-singular matrix $A$:
By creating an augmented matrix with $I_n$ and using row-reduction to transform the original matrix ...
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