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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 matrices are multiplied the way they are multiplied?

I couldn't see a specific reason for multiplying every row of A with every column of B. Is this an arbitrary property of multiplication function of matrices? Instead, why don't we simply multiply row#1 of matrix A with row#1 of matrix B, which would…
user231642
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3 answers

higher dimensional Pauli matrices?

I'm trying to find $3 \times 3$ matrices with some similarity to Pauli matrices. I have some candidates, but they are not perfect. I'm not sure if perfect versions exist, and that is my question. Recall for Pauli matrices $p_k$: $\{p_i,p_j\} =…
Garry
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Non-singular matrix

Assuming $B$, $I+B$, $I+B^{-1}$ are all non-singular, show that $$(I+B)^{-1}+(I+B^{-1})^{-1}=I$$ All I know is that determinants not equal to $0$ and that the inverse of $B$ exists.
Tosh
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similarity transform mapping diagonal matrix of complex conjugates, to real matrix

I know there is some unitary matrix $T$ that maps a diagonal matrix $A = \begin{bmatrix}a+bj & 0 \cr 0 & a-bj \end{bmatrix}$ to a real matrix, namely $$T^{-1}AT = \begin{bmatrix}a & b \cr -b & a \end{bmatrix}$$ but I am having trouble figuring how…
Jason S
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Operator norm increases under taking absolute value of all entries of a matrix

Let $\|A\|:=\sup_{\|v\|=1} \|Av\|$ denote the operator norm induced by the Euclidean distance. If $B$ is a matrix such that $B_{ij} = |A_{ij}|,$ show that $$\|B\|\geq\|A\|.$$
Hedonist
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Are there matrices $A$ and $B$ such that $AB = BA \neq I$

I've been learning about matrices and the identity matrix $I$. It says when $AB = BA = I$, then $A$ and $B$ are inverses of one another. Is it possible for $AB$ to equal $BA$ but not equal $I$?
user99705
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$A+B$ is non singular and $C=(A+B)^{-1}(A-B)$,then prove that $C^TAC=A$

If $A$ is a symmetric and $B$ is a skew symmetric matrix and $A+B$ is non singular and $C=(A+B)^{-1}(A-B)$,then prove that $C^TAC=A$. My…
Vinod Kumar Punia
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$3\times3\times3$ hypermatrix multiplication

Here's an image showing what I am trying to do: The two hypermatrices are multiplied together by taking appropriate slices from each hypermatrix, and realised into a vector by an associated vector, given by (a,b,c) in this case. But I can't figure…
JMP
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Is it true that $\sum_{k=1}^n |a_{kk}| \le \sum_{k=1}^n |\lambda_k|$ for any complex square matrix $A$?

(Note: this is not a duplicate as suggested. I am asking for an inequality.) We know that for any complex square matrix $A$, we have $$\sum_{k=1}^n a_{kk} = \sum_{k=1}^n \lambda_k.$$ I can see this relation in trace property that it is equal sum of…
Ka Wa Yip
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Relation between determinant of the cofactor matrix and the matrix itself?

If the cofactor matrix of A is $$\begin{bmatrix}1 & 2 & 3 \\ 0 & -2 & 4 \\ 0 & 0 & -2\end{bmatrix}$$ How can I find the determinant of $A$?
user276520
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Why we need Invertible Matrices

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or non degenerate) if there exists an n-by-n square matrix B such that $$A B = B A=I_n$$ I know the definition. But what are the practical applications of of…
Layla Smith
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Prove a matrix is invertible

The $2 \times 2$ matrix ${A}$ satisfies $A^2 - 4 {A} - 7{I} = {0},$ where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible. What is the best way to do this?
Rohini Sharma
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Is there relationship between magnitude of matrix-vector multiplication and determinant of that matrix?

If I have a matrix $A$ and vector $x$ is there such a relationship or something similar involving determinants? $$\|Ax\| \leq |\det A|\|x\|$$ where the absolute values indicate the usual Euclidean norm?
AIOM
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Matrix Ambiguity?

I realize that a matrix of objects, where each slot could either be occupied or empty, is impossible to resolve if you are only given the amount of occupied slots in each row and each column. Ex: A $2 \times 2$ matrix with one object in each column…
Johan
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Exponential of a matrix with elements $\cos t \& \sin t$

I want to calculate $e^{A}$ of the matrix $A$: $$\left ( \begin{array}{cc} \cos t & \sin t \\ -\sin t & \cos t \end{array} \right )$$ I tried to use $e^{At}=P\ \mbox{diag}(e^{\lambda t}) P^{-1}$, but from there I obtain the eigenvalue as $\cos…
Xentius
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