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Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra


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  • If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent MatrixIf $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix Let $A$ be a square matrix such that \[A^{\trans}A=A,\] where $A^{\trans}$ is the transpose matrix of $A$. Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.     Hint. Recall the basic properties of transpose […]
  • Sylow Subgroups of a Group of Order 33 is Normal SubgroupsSylow Subgroups of a Group of Order 33 is Normal Subgroups Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.   Hint. We use Sylow's theorem. Review the basic terminologies and Sylow's theorem. Recall that if there is only one $p$-Sylow subgroup $P$ of $G$ for a fixed prime $p$, then $P$ […]
  • If Two Subsets $A, B$ of a Finite Group $G$ are Large Enough, then $G=AB$If Two Subsets $A, B$ of a Finite Group $G$ are Large Enough, then $G=AB$ Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying \[|A|+|B| > |G|.\] Here $|X|$ denotes the cardinality (the number of elements) of the set $X$. Then prove that $G=AB$, where \[AB=\{ab \mid a\in A, b\in B\}.\]   Proof. Since $A, B$ […]
  • Algebraic Number is an Eigenvalue of Matrix with Rational EntriesAlgebraic Number is an Eigenvalue of Matrix with Rational Entries A complex number $z$ is called algebraic number (respectively, algebraic integer) if $z$ is a root of a monic polynomial with rational (respectively, integer) coefficients. Prove that $z \in \C$ is an algebraic number (resp. algebraic integer) if and only if $z$ is an eigenvalue of […]
  • Basis with Respect to Which the Matrix for Linear Transformation is DiagonalBasis with Respect to Which the Matrix for Linear Transformation is Diagonal Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by \[T(ax+b)=(3a+b)x+a+3,\] for any $ax+b\in P_1$. (a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation […]
  • Vector Space of Polynomials and Coordinate VectorsVector Space of Polynomials and Coordinate Vectors Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […]
  • Diagonalize the Upper Triangular Matrix and Find the Power of the MatrixDiagonalize the Upper Triangular Matrix and Find the Power of the Matrix Consider the $2\times 2$ complex matrix \[A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$. (d) Using the result of the […]
  • In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal IdealIn a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$. Show that $P$ is a maximal ideal in $R$.   Definition A commutative ring $R$ is a principal ideal domain (PID) if $R$ is a domain and any ideal $I$ is generated by a single element […]

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