# 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.

35,939 questions
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### Eigenvalues and eigenvectors of reflection matrix

I'm working on the following problem: Let A be a reflection matrix, such that, $a_{ij}=\delta_{ij}-2n_{i}n_{j}$, about a plane perpendicular to $\vec{n}$, $\vec{n}$ being the unitary vector. Find ...
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### Show: $\sum_{i,j=1}^n |a_{ij}|^2=trA^*A=\sum^n_{i=1}\sigma_i^2$

I'm trying to prove the above fact for an arbitrary matrix $A$, with eigenvalue $\lambda_i$, and singular values $\sigma_i$. My approach so far: the trace of a matrix is the sum of its eigenvalues, ...
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### Symmetrical matrix determinant

Is it true that every symmetrical matrix has a determinant non-zero? If so how can I prove it? Note: A symmetrical matrix that is not zero. (thanks to the commenter that pointed it out)
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### Is it possible for an eigenspace to have more than one vector in its basis? What would that imply? Would every vector in the basis be an eigenvector?

So far I've been seeing that the vector that makes up a basis for the null space of the matrix $A-Ix$ (where x is an eigenvalue) is the eigenvector corresponding to the eigenvalue $x$. But I've never ...
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### Is there any matrix $A$ for which $A^{\dagger}A=AA^{\dagger}$ but $A\neq A^{\dagger}$?

I encountered the definition of a normal matrix: $$A^*A = AA^*$$ Whereas the definition of a hermitian matrix is: $$A^{\dagger}=A$$ From this it follows that $A^{\dagger}A=AA^{\dagger}$, very ...
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### Creating a connected undirected graph with connectivity k

I am working on a computing mini-project and I am stuck because of the following problem: "Given a number of nodes N and a connectivity k such that $N > k + 1$, is it always possible to construct ...
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### What does it mean when a matrix has functions as elements and not scalars?

As far as I understand, the matrix $$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix}$$ refers to a linear transformation of the ...
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### Expression for $\|Ax\|_2^2$ in terms of rows of $A$

I am currently reading 1011.3027, Roman Vershynin's introduction to non-asymptotic random matrices. On page 24, there is an equation, (5.24), which reads \begin{align*} \|Ax\|_2^2 = \sum_{i=1}^n\...
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### Unclear proof in the Gauss-Markov theorem

I don't understand a step in the proof of the Gauss-Markov theorem: Mainly, why can we simply replace y with e, given that y is defined as: Thanks in advance!
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### Sum of symmetric, positive semidefinite matrices

Let $A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{p \times n}$. Show that $A^{T}A+ B^{T}B$ is invertible if and only if $\ker A \cap \ker B =\lbrace 0 \rbrace$. I could show that if it's ...
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### Uniqueness of Degenerate Eigenfunctions

Say you have some matrix and the eigenvalues are all the same (degenerate). Let’s say you have $N$ eigenvectors with the label $n$, call them $\psi_n$. If the eigenvalues were not degenerate, then the ...
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I am trying to differentiate over the following Frobenius Norm: $$\Phi =||A-(B\circ C)D ||^2_F$$ with respect to B, C, D respectively, i.e.: $$\frac{\partial \Phi}{\partial B}, \frac{\partial \Phi}{\... 3answers 126 views ### How to solve A^{\frac 12} B A^{\frac 12} = C for A? Suppose that matrices A,B,C are symmetric and positive definite. Then, A has a unique, positive square root, which we call A^{\frac 12}. If$$A^{\frac 12} B A^{\frac 12} = C then can we write ...
Building on top of this question Minimization of Frobenius norm using SDPs, I would like to know for what class of norms ($\Vert \cdot \Vert$ in the following) can we represent the following problem ...
### Is it true that $A^TA=A \implies A^2=A$? Is the converse true?
I was asked this question in an exam. Let $A$ be a square matrix. $A^TA=A \implies A^2=A$, true or false? $A^2=A \implies A^TA=A$, true or false? I rewrote the equations as $(A^T-I)A=0$ and \$(A-I)A=...