# 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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### Let $A=[1 2 0 1 ]$. Find all $2×2$ matrices B, $B≠O_2$ and $B≠I_2$ such that $AB=BA$.

Let $A=[1 2 0 1 ]$. Find all $2×2$ matrices B, $B≠O_2$ and $B≠I_2$ such that $AB=BA$ Explain your answer. I know two ways to find B A) As det of A is 1, it is invertible. So A = B (inverse) B) Or ...
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### Prove the inequality for Condition number of matrix

Let $A \in \mathbb{R}^{n \times n}$ be a non-singular matrix. Let $\hat A=A+\delta A, \ \hat x=x+\delta x, \ \text{and} \ \hat b=b+\delta b$ with $Ax=b$ and $\hat A \hat x=\hat b \$. Here $||.||$...
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### If both $A-B$ and $B-A$ are positive semidefinite, then $A = B$

Let $A, B$ be two positive semidefinite matrices. Prove that if both $A-B$ and $B-A$ are positive semidefinite, then $A = B$. I can show that their diagonal elements are the same but for others I ...
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### Determinant of the Adjoint of a matrix

The determinant of the adjoint of a matrix $A$ is given by $|A|^{n-1}$ where $n$ is the order of the square matrix. So, for odd $n$, the determinant is always positive. Is there an intuitive ...
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### Show that $T_H$ is a linear subspace of $T_G$ so that $\dim H\leq \dim G$

Let $H$ be a subgroup of a matrix group $G$. Show that $T_H$ is a linear subspace of $T_G$ so that $\dim H\leq \dim G$ Definition: Let $\phi:G\rightarrow H$ be a smooth homomorphism of matrix ...
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### Extinction problem for a generalisation of Galton-Watson branching processes

Suppose $A$ is a random $n \times n$ matrix with integer entries in respect to the orthonormal basis $(e_i)_{i=1}^{n} \subset \mathbb{R}^n$. Define $\{v_t\}_{t=1}^{\infty}$ as a sequence of random ...
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### Trace of product of semidefinite matrices is nonnegative

I want to prove this: $A$ is a symmetric positive semi-definite matrix $\Leftrightarrow$ $tr(AB) \geq 0$ $\forall$ B positive semi-definite. I tried using eigenvalues, because they all have to be ...