graphs-of-characteristic-polynomials

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Graphs of characteristic polynomials


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  • Are these vectors in the Nullspace of the Matrix?Are these vectors in the Nullspace of the Matrix? Let $A=\begin{bmatrix} 1 & 0 & 3 & -2 \\ 0 &3 & 1 & 1 \\ 1 & 3 & 4 & -1 \end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$. (a) $\begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}$ […]
  • Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$ Let $A$ be an $n\times n$ nonsingular matrix with integer entries. Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.   Hint. If $B$ is a square matrix whose entries are integers, then the […]
  • Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$ Let $T:\R^3 \to \R^2$ be a linear transformation such that \[ T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 0 \\ 1 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},\] where $\mathbf{e}_1, […]
  • Positive definite Real Symmetric Matrix and its EigenvaluesPositive definite Real Symmetric Matrix and its Eigenvalues A real symmetric $n \times n$ matrix $A$ is called positive definite if \[\mathbf{x}^{\trans}A\mathbf{x}>0\] for all nonzero vectors $\mathbf{x}$ in $\R^n$. (a) Prove that the eigenvalues of a real symmetric positive-definite matrix $A$ are all positive. (b) Prove that if […]
  • Solve a Linear Recurrence Relation Using Vector Space TechniqueSolve a Linear Recurrence Relation Using Vector Space Technique Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be a subspace of $V$ defined by \[U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.\] Let $T$ be the linear transformation from […]
  • Describe the Range of the Matrix Using the Definition of the RangeDescribe the Range of the Matrix Using the Definition of the Range Using the definition of the range of a matrix, describe the range of the matrix \[A=\begin{bmatrix} 2 & 4 & 1 & -5 \\ 1 &2 & 1 & -2 \\ 1 & 2 & 0 & -3 \end{bmatrix}.\]   Solution. By definition, the range $\calR(A)$ of the matrix $A$ is given […]
  • Find a Condition that a Vector be a Linear CombinationFind a Condition that a Vector be a Linear Combination Let \[\mathbf{v}=\begin{bmatrix} a \\ b \\ c \end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}.\] Find the necessary and […]
  • Determine a Matrix From Its EigenvalueDetermine a Matrix From Its Eigenvalue Let \[A=\begin{bmatrix} a & -1\\ 1& 4 \end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is some real number. Suppose that the matrix $A$ has an eigenvalue $3$. (a) Determine the value of $a$. (b) Does the matrix $A$ have eigenvalues other than […]

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