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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.

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How to calculate the inverse matrix?

It is given that, $A$ is $3 \times 3$ matrix and that $$A\begin{pmatrix} x\\y\\z\\\end{pmatrix}= 5\begin{pmatrix} p\\q\\r\\\end{pmatrix}$$ and that $$x= 3p+2q-4r \\ y=p+5q-2r \\ z=7p-6q+3r \\$$ then how can I find $A^\text{-1}$ ?
raf
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Order of products to form System Matrix

Can anyone help explain why does the System Matrix start from the Last process to the first process? Example: Process 1: T1 Process 2: S1 Process 3: T2 Process 4: S2 Process 5: T3 System matrix = product of all process =…
ycY
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Inverse of the sum of a symmetric positive definite matrix and a diagonal

Suppose we have a matrix $XX^T + D$ and we want to calculate its inverse. $D$ is a diagonal matrix. $X$ is a symmetric matrix. We know the inverse of $X_1X_1^T, X_2X_2^T, \ldots, X_nX_n^T$, where $X_1X_1^T, X_2X_2^T, \ldots, X_nX_n^T$ are the…
yx131
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Quasi hermitian $2 \times 2$ matrices

Is there anything significant we can say about matrices of the form $$ \left( \begin{array}{cc} a & b\\ \overline{b} & \overline{a} \end{array} \right) $$ Such a matrix looks almost but not quite like an embedding of the quaternions. Relevance: I'm…
Unwieldy Bob
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If $(AA^t)^r=I$ then $A^tA$ is invertible?

Let $m,n,r$ be natural numbers. Let $A$ be an $m\times n$ matrix with real entries such that $(AA^t)^r=I$, where $I$ is the $m\times m$ identity matrix and $A^t$ is the transpose of the matrix $A$. We can conclude that $m=n$ $AA^t$ is…
dipali mali
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Calculating the product of tridiagonal matrix times its transpose

Denote by tridiag($a$,$b$,$c$) the tridiagonal matrix of size $n \times n$ with diagonal elements $b = (b_1, \ldots,b_n)$. Let $a = (a_1, \ldots, a_{n-1})$ and $c = (c_1,\ldots,c_{n-1})$ be the entries of the other two diagonals. I am trying to…
Stijn
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Prove that another matrix is inverse of $A+BCD$

Given $\mathrm{A} \in \mathbb{F}^{n \times n}, \; \mathrm{B} \in \mathbb{F}^{n \times k}, \; \mathrm{C} \in \mathbb{F}^{k \times k}, \; \mathrm{D} \in \mathbb{F}^{k \times n}$, and $\mathrm{A, \;C}$ are invertible. Prove that $\mathrm{\left(A +…
Shane Hsu
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Matrix and sequences

Given the following matrix : $$ A= \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ \end{bmatrix} $$ and the sequences $a_n$, $b_n$, $ c_n$, $d_n$ such that $ A^n$= \begin{bmatrix} a_n& b_n \\ c_n & d_n \\ …
Lola
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Without directly evaluting, show that the determinant of $A = 0$

Without directly evaluting, show that $det \left[ \begin{array}{ccc} b + c & c + a & b + a \\ a & b & c \\ 1 & 1 & 1 \end{array} \right] =0$ I am stuck on this one. I can only do this by…
Bucephalus
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Show that the "cut norm" on matrices can be made arbitrarily small

Let $M_n(\mathbb{R})$ denote the vector space of matrices of size $n$ with real coefficients. If $M$ is a matrix, let $s(M)$ denote the sum of its elements. We define the "cut norm" of a matrix $A$ in $M_n(\mathbb{R})$ as…
math_lover
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Represent an arbitrary 2x2 matrix entries using sines and cosines

I'm reading Graphics with Mathematica, Fractals, Julia Sets, Patterns, and Natural Forms, by Chonat Getz and Janet Helmstedt. There is a statement: Any matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$, where $a$, $b$, $c$, and $d$ are real numbers, can…
David
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Why is a certain matrix transformation is $\mathbb{R}^2$ to $\mathbb{R}^2$?

Given a matrix transformation $\begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}$, why does it go from $\mathbb{R}^2$ to $\mathbb{R}^2$, when it squishes the plane into a line?
D.R.
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Composite transformation matrix

How to go about calculating the composite transformation matrix? Aren't we supposed to calculate from right to left (just like functions)? If so, why is computation in the picture showing otherwise?
kilitanza
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Let $ A $ be a $ n \times n $ matrix. Show that over the complex numbers

Let $ A $ be a $ n \times n $ matrix. Show that over the complex numbers , there exists an invertible matrix P such that $ P^{-1}AP$ is an upper triangular matrix. Answer: If the matrix $A $ is diagonalisable then $ P^{-1}AP $ is diagonal and…
MAS
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matrix question based on $\omega$ : cube root unity

let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = [p_{ij}]$ be $n\times n$ matrix with $$p_{ij} = \omega^{i+j}$$ Then $P^2$ is not equal to zero where $n$ is equal to $ (a) \ 57 \\ (b)\ 55,58,56 \\ (c)\ 56,48,57 \\ …
Anshi
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