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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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Why are there too many eigenvectors?

The following example is puzzling me. I was given the following matrix: $$\begin{matrix} 0 & 0 & 0 \\ 1/2 & 1 & 0 \\ 1/2 & 0 & 1 \\ \end {matrix}$$ I used an Eigenvector calculator available on the net. I was naive enough to plug my matrix in…
zoli
  • 20,452
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Powers of $2\times 2$ matrices, such that $A^n = I$

For what $n$ natural number does there exist a real $2\times 2$ matrix $A$, such that $A^n = I$? $n=2,3$ clearly works, because $(-I)^2 = I$, and for $n=3$ we have $\left( \begin{smallmatrix} -2 & 1\\ -3 & 1 \\\end{smallmatrix} \right)$ However, I…
Jake1234
  • 1,169
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How to measure the similarity of a matrix to triangular form

Given a matrix A, how to measure its similarity (row and column permutations can be performed on A) to triangular form, here the triangular form is like \begin{align} M=\begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1…
jiazhen he
  • 31
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Normal subgroup proof

I'm preparing for an algebra exam later this month and am trying out the exercises from my textbook. Sadly I got stuck with this one: Let $G$ be a group of all regular upper triangular matrices $2 \times 2$ over $\mathbb{Q}$. Let $H$ be its subgroup…
Ant150
  • 57
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Efficient inversion of a symmetric, positive definite matrix

I have to invert a symmetric, positive definite matrix in order to execute an extended Kalman Filter. I know quite some matrix decompositon methods like Cholesky or QR, but the question is what is the most efficient way doing it for this specific…
BThomas
  • 133
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$A^TCA \leq B^TCB \Rightarrow A^TA \leq B^TB$?

Let $A$,$B \in \mathbb{R}^n$, $C\in\mathbb{R}^{n\times n}$, and $C=D^TD$ where $D$ is a $n\times n$ psd matrix, is it guaranteed that $A^TCA \leq B^TCB \Rightarrow A^TA \leq B^TB$?
thinkbear
  • 391
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Creating a zero matrix through matrix multiplication

From an assignment: Let $A = \left[ \begin{matrix} 3 & -6 \\ -2 & 4 \end{matrix}\right] $ Construct a $2 * 2$ matrix $B$ such that $AB$ is the zero matrix. Use two different nonzero columns for $B$. The value of $AB$ would be: $$ AB =…
user17404
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1 answer

What is known about totally positive matrices?

A totally positive matrix is one whose minors are all positive. This is a simple elementary concept but most of the development on the subject is far from elementary. I am having a hard time understanding most papers on the subject because of the…
wircho
  • 1,431
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2 answers

Find the values of $k$ so that the matrix is not invertible

So, the question is indeed asking for what you just read. I have the following matrix for which I have to find the values of $k$ in order to make it not invertible. I have understood that the inverse of a matrix is the result of $[A | I]$, so since…
Irais
  • 31
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find the solution for $h^{T} \Phi^{2k} h = \delta_{k}$

I'm trying to find the general expression for $h$, an $n \times 1$ vector, which solves $$h^{T} \Phi^{2k} h = \delta_{k} $$ where $k$ is a non-negative integer, $\Phi$ is an $n \times n$ diagonalizable matrix (assume $\Phi$ is full rank) and…
user1147431
  • 21
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$0\times n$ matrices

I think it is desirable to have that $M_{m\times n}\left(\mathbb{K}\right)\not=M_{m'\times n'}\left(\mathbb{K}\right)$ if $m\not=m'$ or $n\not=n'$. In other words, the set of all $m\times n$ matrices on $\mathbb{K}$ should be different from the set…
user112679
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1 answer

Why is there two versions of the rotation matrix?

Why is there two versions..for example, I got some matrices for the x axis rotation This is for the X AXIS the other one is for the x axis also, it is.. I think it could be from going counterclockwise and clockwise(right hand rule?) on the…
terry
  • 71
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Is the null space inside the collumn space of a matrix?

From what I've seen online, it seems that the null space isn't in the column space, but I don't understand why that is the case. If the null space is the set of all combinations that equal 0, and the column space is the set of all combinations,…
tokola
  • 697
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if matrix multiplication $B*A=C*A$, does it mean $B=C$?

If matrix multiplication $B*A=C*A$, does it mean $B=C$? If A is invertible, then I guess this should work. If not, then?
CyberPlayerOne
  • 617
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Show that $A$ is invertible and that $AB=BA$

Let $A,B\in\mathcal{M}_n(\mathbb{R})$. We suppose that there exists $P\in\mathbb{R}[X]$ with $\deg P\geq1$ such that $P(0)=1,P(A)=AB$. How can I show that $A$ is invertible and that $AB=BA$ ? I don't really see how to exploit the information given…
Hippalectryon
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