# 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.

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### Why is the solution set of the reduced row echelon form of A equal to the solution set of A?

One way of solving a system of linear equations is to express it in an augmented matrix. Then, we can perform elementary row operations in order to bring the matrix into RREF (reduced row echelon form)...
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### If matrix $A=[a_{ij}]_{4 \times 4}$ such that…

If matrix $A=[a_{ij}]_{4 \times 4}$ such that $a_{ij}= \begin{cases} 2&\text{if}\, i=j\\ 0&\text{if}\, i \not= j\\ \end{cases}$ , then $\{\frac{det(adj(adjA))}{7}\}$ is ($\{x\}$ ...
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### Multiplication by an extrinsic matrix

I need to project a 2D point taken by a Kinect camera to real world coordinates, without using the official Kinect SDK. I found this answer: https://stackoverflow.com/questions/47348266/color-space-to-...
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### Showing a matrix identity

Let $A$ and $B$ be symmetric, positive definite matrices. Consider the following equations: $$B = XAX, \qquad A = YBY,$$ where $X$ and $Y$ are also symmetric and positive definite matrices. These ...
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### Defective Matrices Transforms

Defective Matrix:$\begin{bmatrix}1&1\\0&1\end{bmatrix}$ I am unable to wrap my head around what kind of transforms yield defective matrices or matrices where geometric multiplicity is less ...
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### Strong separation of symmetric rank $1$ matrices

Let $V$ be the inner product space of $n \times n$ symmetric matrices where the inner product is given by $\langle A, B \rangle = trace(AB)$ and let $A, B \in V$ be such that they have eigenvalue $1$ ...
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### Same Eigenvalues = Similar Matrices?

While watching Strang's lecture on similar matrices he stated that if any two matrices,$A$ and $B$, have the same eigenvalues then they can be put in the form $A=P B P^{-1}$ . This is very easy to see ...
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### If $\operatorname{Tr}(H)\ge 0$, is $\operatorname{Tr}(PH)\ge 0?\$ $H$ is Hermitian, $P$ is positive definite.

Old: Given $\operatorname{Tr}(DH)\ge 0$ for positive real diagonal $D$ and Hermitian $H$, is $\operatorname{Tr}(H)\ge 0$? Let $D$ be positive real diagonal and $H$ be Hermitian such that $H$ may have ...
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### There does not exist a onto ring homomorphism from $M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F)$ for any field $\mathbb F.$

For a positive integer $n$, I have to show that there does not exist a onto ring homomorphism from $M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F)$ for any field $\mathbb F.$ If it ...
I have the following equation $p=(1+r)(pA+wl)$ where w, r are scalars; p, l are vectors; A is a matrix. I want to prove that an increase in w would lead to a decrease in A or l (i.e., p is constant). ...