Mathematics Stack Exchange
Mathematics Stack Exchange

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.

A matrix is a rectangular array of elements, usually numbers or variables, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has $m \times n$ elements and is called an $m$ by $n$ matrix. Matrices are a part of .

Matrices can be added and subtracted. Furthermore, if they have compatible shapes, they can be multiplied. More precisely, given two matrices $A$ and $B$, the matrix $AB$ is defined when the number of columns of $A$ is equal to the number of rows of $B$. In particular, given a natural number $n$, any two matrices $A$ and $B$ with $n$ columns and $n$ rows can be multiplied in both ways (that is, both $AB$ and $BA$ exist).

For questions specifically concerning matrix equations, use the tag.

55954 questions
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How would I reduce my matrix even further?

How would I reduce my matrix even further?
Megan Reynolds
  • 191
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Let $L$ denotes the set of all primes $p$ such that the following matrix is invertible

I am stuck on the following problem: Let $L$ denotes the set of all primes $p$ such that the following matrix is invertible when considered as a matrix with entries in $\Bbb Z/p \Bbb Z$ . $A=\begin{pmatrix} 1 &2 &0 \\ 0 &3 &-1 \\ -2 &0 …
learner
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How to prove $A+ A^T$ symmetric, $A-A^T$ skew-symmetric.

Prove that if $A$ is a square matrix, then: a) $A+ A^T$ is symmetric. b) $A-A^T$ is skew-symmetric. c) Use part (a) and (b) to show $A$ can be written as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, with $A = B + C$.
Kyla
  • 73
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differentiate a matrix trace

Let $A,B,X$ be square matrices of the same size. I know $\ \displaystyle \frac{\mathrm{d}~ \mathrm{trace} (AX)}{\mathrm{d}X}=A.\quad$ Is it true that $\displaystyle \frac{\mathrm{d}~ B~ \mathrm{trace} (AX)}{dX}=B\otimes A$, where $\otimes$ means…
Sunni
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How to solve this system of linear equations

$$M = \left(\begin{smallmatrix} a_1 & a_2 & a_3 & a_4\\ b_1 & b_2 & b_3 & b_4\\ a_1 & c_2 & b_2 & c_4\\ a_4 & d_2 & b_3 & c_4\\ b_1 & c_2 & a_2 & e_4\\ b_4 & d_2 & a_3 & e_4\end{smallmatrix}\right)$$ All of the equations equal to 26; augmented, the…
Don Larynx
  • 4,703
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Solving a Matrix Equation for a Matrix

I have a transformation that is determined by the multiplication of several matrices: Transformation = $\mathbf{ABC}$. I need to find such matrix $\mathbf{K}$ to satisfy equation: $\mathbf{ABC} = \mathbf{ACKB}$ .
Ivan
  • 123
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If one of the sub-matrices is singular, the whole matrix will be singular?

A larger matrix is formed by four sub-matrices A,B,C,D. If one of the sub-matrices is singular, the whole matrix will be singular?
user93019
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How can you tell how many invariant factors a matrix has?

In my notes, I have a $4\times 4$ complex matrix $A$ with the following properties. The characteristic polynomial of $A$ is $(x-1)(x+1)^3$, and the geometric multiplicity of $-1$ is $2$. That is all that is known about $A$. In determining the…
Corbin
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Constant in the positive-definite condition of matrices

Suppose I have a positive definite matrix $A$, so $$x^TAx \geq C_A|x|^2$$ holds for all $x$. $A^{-1}$ is positive definite too: $$y^TA^{-1}y \geq C_B|y|^2,$$ is there any way I find the constant $C_B$ in terms of $C_A$?
studentX
  • 159
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Row Reduction of a matrix

Let us consider the following matrix $$M= \left[ {\begin{array}{cc} 1 & 1 & 2 & 1 & 5 \\ 1 & 1 & 2 & 6 & 10 \\ 1 & 2 & 5 & 2 & 7 \\ \end{array} } \right]$$ I was able to reduce the above matrix…
amir
  • 1,311
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2 answers

Normal matrices

Does $\|Tv\|\leq\|v\|$ (for all $v \in V$) leads to $T$ is normal? If not, when I add the additional information that every e.e of $T$ is of the absolute value 1, can I prove $T$ is unitary? Thanks!
ORBOT Inc.
  • 907
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Sum preserving transformations of square symmetric matrices with natural elements

Suppose I have a matrix $A = [a_{ij}]_{n\times n}$ such that $a_{ij} \in \mathbb{N}$ and $a_{ij}=a_{ji}$. I want to transform this matrix such that the sum $\sum_{i,j}a_{ij}$ remains constant, symmetry is maintained, and the resultant matrix also…
ananta
  • 431
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Linear transformation? Image?

Consider the space of functions $F=\operatorname{span}\lbrace \sin(x),\cos(x),x\cdot \sin(x), x\cdot \cos(x)\rbrace$ and the image $L:F\rightarrow F:f\rightarrow -f''-2f$. Show that $L$ is a linear image. Find the matrix $M$ of the linear images…
am87gu
  • 101
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Matrix Identity

Let $A$ be a $ n \times n $ positive definite matrix, $P$ be a $n\times m$ real matrix with full column rank, I'm wondering whether we have the following inequalty $$ P( P^{T} A P) ^{-1} P^{T} \preceq A ^{-1} ,$$ And in which sufficient conditions…
mewmew
  • 483
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4 answers

Associativity of Matrix Multiplication

Let $A$, $B$, and $C$ be vectors. Suppose $A$ is $1 \times 10$, and $B$ is $10 \times 1$ and $C$ is $5 \times 1$. Then if we multiply $(AB)C$, the result is a $5 \times 1$ vector, but if we try to multiply $A(BC)$, the dimensions are not…
user25004
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