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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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A matrix equation to solve

Let $C$, $p_1,\ldots,p_n$ and $q_1,\ldots,q_n$ be positive real numbers. How solve this equation in general to end up with a general formula for each $x_k$? \begin{equation} \left(\begin{array}{cccccc} p_1 & p_2 & p_3 & \cdots & p_{n-1} &…
hardy5
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What's the insight for a 3x3 matrix with orthogonormal columns,the rows are also orthogonormal?

I know this can be easily proved with simple matrix tricks, But I don't know the insight for this, and just feels it amazing that if I pick up 3 orthogonormal vectors in 3d space, their corresponding x,y,z portions automatically forms orthogonormal…
user2559257
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Find matrix commuting with all projections

Please help me solve this problem: Find matrix $A$ such that $A$ commutes with all projection matrix (A matrix $P$ is said to be projection if $P^2 = P$). Thank you so much.
user97656
2
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1 answer

Linear Algebra Matrix algebra

Let $S: \mathbb R^3 \rightarrow \mathbb R$ $v= (v_1,v_2,v_3) w= (w_1,w_2,w_3)$ Both w and v are vectors Express the standard matrix $S: \mathbb R^3 \rightarrow \mathbb R$ in terms of $v_1,v_2,v_3$ and $w_1,w_2,w_3....$ I have trouble…
Chris Kim
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Question on Matrix Transform Operations

Given $e = Y - XB$, where $ e = \begin{bmatrix} e_1 \\ \vdots \\ e_n \\ \end{bmatrix} $, $ Y= \begin{bmatrix} y_1 \\ \vdots \\ y_n \\ \end{bmatrix} $, $ X= \begin{bmatrix} 1 & x_1 & x^2_1 \\ \vdots & \vdots & \vdots \\ 1 & x_n &…
KeithSmith
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Using Rref to find the inverse of a matrix.

Since, I can't divide vectors to deduce an inverse matrix I have dismissed that approach. I did find that if I multiply all of the matrix row operators It will yield the inverse. Since I did the logic work to put my original matrix into RRef. I can…
Chris
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comparison between matrices

Let $M_1$ and $M_2$ be two symplectic matrices of dimension $2n_1\times 2n_1$ and $2n_2\times 2n_2$ respectively; Let $P_1>0$ and $P_2>0$ be the positive definite solutions of the following equations $$ \left[\begin{array}{cc} I_{n_1} &…
sid
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Kronecker delta matrix question

So I have the Kronecker delta which is denoted as $\delta_{ij}$=$I$. Let $a_1, a_2, \cdots, a_n$ be a set of $n$ real numbers, I must show that: $\sum\limits_{i=1}^n a_i \delta_{ij} = a_j$ and $\sum\limits_{j=1}^n a_j \delta_{ij} = a_i$ i.e.…
Ryan
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Regularization of a matrix

Why does adding a value to the diagonal of the matrix stabilise it? Why not add the value to all elements? So, before I use Cholesky to find the x in Ax=b, I was told to add a regularisation paramter to the diagonal of my A.
user1876942
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Multiply a matrix by its transpose

I'm using some material found on the internet for learn how to use R. One exercise is asking to return the result of a multiplication of a $15\times 3$ matrix by its transpose. Being the transpose a $3\times 15$ matrix, I would assume the result is…
Gianluca
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system of equation (3 unknown, 3 equations)

I want to solve a system of equations, but I seem to get it wrong. Problem: see picture, and note that I should tell for which a I have no solution one solution $\infty$ solutions Attempt: $$\begin{matrix}i \\ ii \\…
jacob
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Rank of a matrix with structure

Let $P \in [0,1]^{n \times n}$ ($n > 1$) be a matrix such that the diagonal entries $P_{ii} ~~\forall i$ are $0$ and upper diagonal entries $P_{ij} ~~\forall i < j$ $\in (0.5,1)$ and lower diagonal entries $P_{ij} ~~ \forall i > j$ $\in (0,0.5)$.…
Ved
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If $a^HUa=0$ for all $a$, can we conclude that $U=0$?

I have the following equation: $a^HUa=0$ where '$a$' can be any arbitrary vector and $U$ is a matrix ($H$ means Hermitian). Can we conclude that $U=0$? Any reference? Thanks.
user99866
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What is the result of $x^{\top}A\dot{x}+\dot{x}^{\top}Ax$?

What is the result of $x^{\top}A\dot{x}+\dot{x}^{\top}Ax$, provided that $A=A^{\top}$? Actually, I wanted to expand $d(x^{\top}Ax)\over{dt}$.
Ansu
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What is the name of matrices of this form?

Consider the upper traingular $N \times N$ matrix $$\left(\begin{array}{cccccccc} 0 & b_{1} & \dots & b_{q} & 0 & 0 & \dots & 0\\ \vdots & 0 & b_{1} & \dots & b_{q} & 0 & \dots & 0\\ \vdots & & \ddots & & & & & \vdots\\ \vdots & & & 0 & b_{1}…
rwolst
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