# Tagged Questions

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### How can I calculate matrix differentiation? [duplicate]

I am studying about the Matrix Differentiation. I don't know if this red box differential metric, which is how it is calculated.
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### Determinant-like expression for non-square matrices

I'm interested in whether for any real matrix of size $m \times n$ there is a real number with the following properties: It is a polynomial expression with real coefficients in the entries of the ...
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### Solve linear equations [on hold]

\begin{bmatrix} 0 & 0& 1& 1& 1&0 \\ 0 & 0& 0& 0& 2&1 \\ -3 & 0 & 0 & 0 & -2& 0\\ 4& 4& 0& 0& 0 & -1\\ ...
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### Why we use $\mathbb{R}^{m \times n}$ notation instead of $\mathbb{R}^{n \times m}$?

I just realised, that I use all the time the notation $\mathbb{R}^{n \times m}$, and all books and papers use $\mathbb{R}^{m \times n}$. $\mathbb{R}^{n \times m}$ is more sympathetic for me, because I ...
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### Is the basis of null space of a matrix always a subset of the basis of its column space?

Given an $m\times n$ matrix $A$, is the basis of its null space (set of $x$ such that $Ax=0$) always a subset of the basis of the row space of $A$? In general, the basis of a subspace may not be a ...
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### Show that U subspace is supplementary to the kernel. How to find values of a b c d using intersection of two matrices.

I already found the kernel to be \begin{pmatrix} -2c&-2d\\c&d \end{pmatrix}. and U is a subspace of a $M_2$ matrix defined by \begin{pmatrix} a&b\\2a&2b \end{pmatrix}. So i have to ...
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### determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
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### Any article expounding the difference between matrix analysis and functional analysis?

I do theoretical physics. For quantum mechanics, the mathematical foundation is rigorously functional analysis. However, people generally take matrix analysis (for finite dimensional vector spaces) to ...
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### Families of Square Roots of Identity Matrices

I just analysed this equation for real matrices $$A^2=\begin{pmatrix}a&b\\c&d\end{pmatrix}^2=I$$ From the main diagonal of $A^2$ we must have $a^2+bc=bc+d^2=1$ showing that $d=\pm a$. CASE ...
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### What does $\langle x,X\rangle$ mean?

I encountered lots of times but can't find the meaning of $\langle x,X\rangle$. I know in general $\langle a,b\rangle$ is the inner product of two vectors, but this is obscure. I'm talking about real ...
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### Let $A$ be an $n\times n$ invertible complex matrix such that $A^7 = A^*$. Show that $A^8 = I$.

Let $A$ be an $n\times n$ invertible complex matrix such that $A^7 = A^*$ (where $*$ denotes conjugate transpose). Show that $A^8 = I$. Here are my thoughts so far: I was able to show that all ...
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### Is there any restriction to the sum of eigenvalues for non-negative, irreduceble and square matrices?

I'm trying to find if there is a restriction in tr(A) or eigenvalues sum for a non-negative, irreducible square matrix A. As an additional information, the row sums and the order of the matrix is ...
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### Mutually commuting matrices

Let $A_{1},..., A_{m}$ be $n \times n$ matrices with entries in a field $K$ such that $A_{i}A_{j} = A_{j}A_{i}$ for all $1 \leq i, j \leq n$ and the product $A_{1}A_{2} ... A_{m} = 0$ is the zero ...
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### How to get the transformation matrix for Linear Discriminant Analysis?

I am trying to implement Linear Discriminant Analysis. Is the eigen vectors of the product of within scatter matrix and between scatter matrix inverse (Sw*Sbinverse), the transformation matrix? Could ...
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### non-symmetric matrix with orthogonal eigenvectors

Given that a symmetric matrix with real entries has orthogonal eigenvectors, is the converse true? That is, if a matrix has orthogonal eigenvectors, does it have to be symmetrical and real?