# symmetric-matrix

• If the Order is an Even Perfect Number, then a Group is not Simple (a) Show that if a group $G$ has the following order, then it is not simple. $28$ $496$ $8128$ (b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple. Hint. Use Sylow's theorem. (See the post Sylow’s Theorem […]
• A Condition that a Vector is a Linear Combination of Columns Vectors of a Matrix Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$. Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the […]
• A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}$ is diagonalizable. Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix. That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]
• Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$ Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings. (a) $\rk(AB) \leq \rk(A)$. (b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.   Hint. The rank of an $m \times n$ matrix $M$ is the dimension of the range […]
• Quotient Group of Abelian Group is Abelian Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$. Then prove that the quotient group $G/N$ is also an abelian group.   Proof. Each element of $G/N$ is a coset $aN$ for some $a\in G$. Let $aN, bN$ be arbitrary elements of $G/N$, where $a, b\in […] • Group Homomorphism from$\Z/n\Z$to$\Z/m\Z$When$m$Divides$n$Let$m$and$n$be positive integers such that$m \mid n$. (a) Prove that the map$\phi:\Zmod{n} \to \Zmod{m}$sending$a+n\Z$to$a+m\Z$for any$a\in \Z$is well-defined. (b) Prove that$\phi$is a group homomorphism. (c) Prove that$\phi$is surjective. (d) Determine […] • Transpose of a Matrix and Eigenvalues and Related Questions Let$A$be an$n \times n$real matrix. Prove the followings. (a) The matrix$AA^{\trans}$is a symmetric matrix. (b) The set of eigenvalues of$A$and the set of eigenvalues of$A^{\trans}$are equal. (c) The matrix$AA^{\trans}$is non-negative definite. (An$n\times n$[…] • Complement of Independent Events are Independent Let$E$and$F$be independent events. Let$F^c$be the complement of$F$. Prove that$E$and$F^c$are independent as well. Solution. Note that$E\cap F$and$E \cap F^c$are disjoint and$E = (E \cap F) \cup (E \cap F^c)\$. It follows that \[P(E) = P(E \cap F) + P(E […]