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Questions tagged [matrix-analysis]

For question about matrices and their algebraic properties. Together with [tag:linear-algebra] if necessary.

In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties.

Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).

See also: Wikipedia

272 questions
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How to find out a smallest sub-matrix B from a sparse matrix A which has the equal rank and # of non-zero columns?

I have a very sparse matrix A. I need to find out a smallest number of rows of A which satisfies the following conditions: 1). Let us suppose the number of rows form a sub-matrix B. In another word, for a given sparse tall matrix A, we need to find…
Benson
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Diophantine matrix equation - Solver or algorithm

I look for a solver or reference rather than an answer about how to solve the diophantine matrix equation. One states it below. $\mathbf{X}\mathbf{N} + \mathbf{Y}\mathbf{D} = \mathbf{I}$ All entries of X, N, Y and D belongs to polynomials.…
Bruno Lobo
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Show that $B \in \Bbb M_3 (\Bbb Z).$

Let $A \in \Bbb M_3 (\Bbb Z)$ be such that $A=B^2,$ for some $B \in \Bbb M_3 (\Bbb R).$ Show that $B \in \Bbb M_3 (\Bbb Z).$
math maniac.
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Rational canonical form conjugation

I know that if A is an nxn intger matrix, with A = XRX^-1 where R is the rcf of A. Then R is also an integer matrix. My question is Does X and X^- are integer matrices as well ?
user593805
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Rational canonical form of integer matrix

In the paper Computing Rational Forms of Integer Matrices by MARK GIESBRECHT† AND ARNE STORJOHANN It says that, When A ∈ Qn×n has all integer entries, the Frobenius form F of A has all integer entries as well. But how about if A ∈ Znxn is…
user593805
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Rational canonical form of some square of a matrix

Let $$ B = \begin{bmatrix} -2 & 0 & 0 \\ -1 & -4 & -1 \\ 2 & 4 & 0 \\ \end{bmatrix} $$ Then $$ B^2 = A = \begin{bmatrix} 4 & 0 & 0 \\ 4 & 12 & 4 \\ -8 & -16 & -4 \\ \end{bmatrix} $$ But the square of rcf of B is not equal to rcf of A. Why is…
user593805
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Eigenvetor Property of a Matrix

If a matrix $A$ is complex orthogonally similar to an upper triangular matrix, that is, $A=QUQ^T, Q^TQ=I$ and $U$ is upper triangular matrix, then there exist at least one eigenvector $x$ of $A$ such that $x^Tx\neq 0.$ This is an exercise in Horn…
user602672
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Let A,B be nxn matrix If $(A^2)(B^2) = (B^2)(A^2)$ , is $AB=BA$?

Let $A,B$ be $n\times n$ matrix If $(A^2)(B^2) = (B^2)(A^2)$ , is $AB=BA$?
user593805
  • 185