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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.

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Eigenvalues of $A$ and $A + A^T$

This question has popped up at me several times in my research in differential equations and other areas: Let $A$ be a real $N \times N$ matrix. How are the eigenvalues of $A$ and $A + A^T$ related? ...
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Why is the Operator Norm so hard to calculate?

I recently took a better look at the operator norm defined on a matrix $\mathbf A \in \Bbb{K}^{n\times n}$ as follows: $$\|\mathbf A\|_p=\sup\{\|\mathbf Ax\|_p \mid x\in\Bbb{K}^n\land\|x\|=1\}$$ ...
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sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent Gaussian random variables. The trouble is, my Gaussian random variables are not independent. ...
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Understanding rotation matrices

How does ${\sqrt 2 \over 2} = \cos (45^\circ)$? Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function represent? ...
30k views

Matrices: left inverse is also right inverse? [duplicate]

If $A$ and $B$ are square matrices, and $AB=I$, then I think it is also true that $BA=I$. In fact, this Wikipedia page says that this "follows from the theory of matrices". I assume there's a nice ...
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Sudoku with special properties

Sudoku is a puzzle, with the objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid (also "sudoku-blocks") contains all of ...
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A surprising result about the product of Blaschke matrices

I have verified analytically the conjecture described bellow up to $n=4$, but have had no success trying to prove it. Any help would be much appreciated. Setup Let $\{\lambda_i\}_{i=1}^n$ be real ...
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If $A$ and $B$ are matrices such that $AB^2=BA$ and $A^4=I$ then find $B^{16}$.

If $A$ and $B$ are matrices such that $AB^2=BA$ and $A^4=I$, then find $B^{16}$. My Method: Given $$AB^2=BA \tag{1}$$ Post multiplying with $B^2$ we get $$AB^4=BAB^2=B^2A$$ Hence $$AB^4=B^2A$$ Pre ...
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Are $10\times 10$ matrices spanned by powers of a single matrix?

I don't know how to answer this question: Is there a $10 \times 10$ matrix $A$ such that $$M_{10}(\mathbb{F})=\text{span}\{I,A,A^2,\ldots, A^{100}\}\textrm{,}$$ where $M_{10}(\mathbb{F})$ is the ...
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Suppose $M$ is an $m \times n$ matrix such that all rows and columns of $M$ sum to $1$. Show that $m=n$

I find this a rather awkward question, and I was given a hint: use invariants, which I found even more awkward. Suppose $M$ is an $m \times n$ matrix such that all rows and columns of $M$ sum to $1$...
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What kind of matrix is this and why does this happen?

So I was studying Markov chains and I came across this matrix \begin{align*}P=\left( \begin{array}{ccccc} 0 & \frac{1}{4} & \frac{3}{4} & 0 & 0\\ \frac{1}{4} & 0 &...
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Sylvester's determinant identity

Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$\det(I_m+AB) = \det(I_n+BA)$$ where $I_m$ and $I_n$ denote the $m \times m$ ...
37k views

Inverse of a Positive Definite

Let K be nonsingular symmetric matrix, prove that if K is a positive definite so is $K^{-1}$ . My attempt: I have that $K = K^T$ so $x^TKx = x^TK^Tx = (xK)^Tx = (xIK)^Tx$ and then I don't know ...
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What is the theory of Matrices?

I've just passed high school and studying matrices. I've learned about determinant, transpose and adjoint etc. I've learned the method of finding these things but what's the purpose of finding these ...
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Traces of all positive powers of a matrix are zero implies it is nilpotent

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be $0$, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like ...
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What is an interpretation of the matrix exponential?

I just read about the existence of the "matrix exponential" $$e^X := \sum_{k = 0}^\infty\frac1{k!}X^k$$ Is there a simple way to interpret this? I understand the analog between real number ...
27k views

Elegant proofs that similar matrices have the same characteristic polynomial?

It's a simple exercise to show that two similar matrices has the same eigenvalues and eigenvectors (my favorite way is noting that they represent the same linear transformation in different bases). ...
37k views

The transpose of a permutation matrix is its inverse.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
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How to show path-connectedness of $GL(n,\mathbb{C})$
Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...