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.

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Matrix Polynomial with Non-Zero Coefficients

Let $A ∈ M_{n \times n}(F)$ for some field $F$. Prove there is some non-zero polynomial $p(x)$ such that $p(A) = 0$. I approached this question by dealing with each entry of the matrix and equating it to the respective $0$ on the $0$ matrix but I am…
Rij
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Write $M=(e^{a_{i,j}})$ as a function of $A=(a_{i,j})$

I know that this matrix $$M=\begin{pmatrix} e^{a_{1,1}} & e^{a_{1,2}} & e^{a_{1,3}}\\ e^{a_{2,1}} & e^{a_{2,2}} & e^{a_{2,3}} \\ e^{a_{3,1}} & e^{a_{3,2}} & e^{a_{3,3}} \end{pmatrix}$$ can't be expressed as $\exp(A)$…
Mark
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Combining Matrix System of equations

This is a question I came across while reviewing: I'm looking for some guidance on how to approach it I have the system of matrix equations below: C*z + transpose(A)y = f A*z = g Express this problem as a matrix equation Mx = b for suitably defined…
bhenry
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Matrix with entities 0 and 1

The number of $3\times 3$ matrix A whose entities are either 0 or 1 and for which the system $A\left[ {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ 0\\ 0 \end{array}} \right]$ has exactly two distinct…
Samar Imam Zaidi
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Meaning behind multiplication of matrix and its transpose

When you multiply a matrix $M$ by its transpose, what exactly does this product represent? What does each entry represent? I see that a lot of these examples, when a document-term matrix (DTM) is created and then this DTM is multiplied by its…
sanjeev
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nonsingularity of diagonally dominant matrix

I have a nonnegative matrix which is sort of diagonally dominant but not quite. I have $a_{ii}>a_{ij}$ and $a_{ii}>a_{ji}$ for any $i$ and $j$ not equal to $i$. I wonder whether it must be nonsingular.
lioqq
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any monic polynomial of degree $n$ whose root is $A$ is the characteristic polynomial of $A$?

Consider any matrix $A$ of order $n\times n.$ How to show that any monic polynomial of degree $n$ whose root is $A$ is the characteristic polynomial of $A?$ I have used the result several times without knowing the proof.
Sriti Mallick
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Find all the least squares solutions of Ax=b

Find all least squares solutions of A x = b, where A = \begin{bmatrix} 1 & 3 \\[0.3em] -2 &-6 \\[0.3em] 3 & 9 \end{bmatrix} and b = \begin{bmatrix} 1 \\[0.3em] 0 …
user51462
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Positive defnite matrices

Hei guys, I would like to check with you a prove I have Let the matrix $X_1 = \begin{bmatrix} A_1 & B_1\\ B_1 ^ T & A_1\\ \end{bmatrix}$ be positive definite. Based on this I would like to show that the matrix $X_2 = \begin{bmatrix} …
Bogdan
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How to show that the general orthogonal group $O(n,1)$ has four components?

Define a bilinear form on $\mathbb R^{n+1}$ by $(x,y)=\sum\limits_{i=1}^n x_iy_i-x_{n+1}y_{n+1}$, the general orthogonal group $O(n,1)$ is defined to be $\left\{B\in GL(n+1,\mathbb R)\mid(Bx,By)=(x,y),\forall x,y\in\mathbb R^{n+1}\right\}$, where…
user75622
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Let $A = \begin{pmatrix} a & b & c \\ 0 & a & b \\ 0 & 0 & a \\ \end{pmatrix} $. Find $A^n$

I am new here so please let me know if I must resentence the exercice. I considered it too short not to include it in the title too. Let $ A= \begin{pmatrix} a & b & c \\ 0 & a & b \\ 0 & 0 & a \\ \end{pmatrix}$ Find $A^n$ I…
Dareal1 62
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I'm given a matrix equation $XA = X - A$, where I know what $A$ is, how do I solve such a thing?

Here's the matrix $A$: \begin{pmatrix} -1 & 1 & 2\\ 2 & -2 & 3\\ -1 & 1 & -2 \end{pmatrix} How do I solve the matrix equation $XA = X - A$?
Ognjen Ognjanovic
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Complex square matrices $A, B$ where $A$ and $B$ have a square root but $A+B$ doesn't

Are there any matrices like this? I think yes... choose $B$ to be all zeros except $1$ at an off diagonal component $(i,j)$ that affords $B$ a square root. Then choose $A$ to be a normal matrix where adding $1$ to the $(i,j)$ component makes it…
Christian Chapman
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Two dimensional rotation of a tensor component UV

I have a quantity $[UV]$ where the $[\hspace{5pt}]$ represents ensemble averaging and $U, V$ are the velocity components in $X$ and $Y$ directions. In a new reference frame (say: $x, y$), which is rotated by an angle "$a$" in the CLOCKWISE direction…
user77153
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Circular shift and reflection

I know that in a circulant matrix(C) all other rows are (right) shifted versions of first row. Let $x = [c_{0},c_{N-1},c_{N-2},...,c_{1}]$ be the first row. Then $x(n-j) = [c_{j},c_{j-1},c_{j-1},...,c_{j+1}]$ will be the $j^{th}$ row, where $j =…
Vinod
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