# field-theory-eyecatch

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• Conditions on Coefficients that a Matrix is Nonsingular (a) Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that the entries of the matrix $A$ satisfy the following relation. $|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|$ for all $1 \leq i \leq n$. Show that the matrix $A$ is nonsingular. (b) Let […]
• How to Find Eigenvalues of a Specific Matrix. Find all eigenvalues of the following $n \times n$ matrix. $A=\begin{bmatrix} 0 & 0 & \cdots & 0 &1 \\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 &0\\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & […] • The Coordinate Vector for a Polynomial with respect to the Given Basis Let \mathrm{P}_3 denote the set of polynomials of degree 3 or less with real coefficients. Consider the ordered basis \[B = \left\{ 1+x , 1+x^2 , x - x^2 + 2x^3 , 1 - x - x^2 \right\}.$ Write the coordinate vector for the polynomial $f(x) = -3 + 2x^3$ in terms of the basis […]
• Matrices Satisfying the Relation $HE-EH=2E$ Let $H$ and $E$ be $n \times n$ matrices satisfying the relation $HE-EH=2E.$ Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then […]
• Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors Let $V$ be a vector space and $B$ be a basis for $V$. Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$. Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, […] • A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues Let$A$be an$n\times n$real symmetric matrix whose eigenvalues are all non-negative real numbers. Show that there is an$n \times n$real matrix$B$such that$B^2=A$. Hint. Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix. […] • An Example of a Matrix that Cannot Be a Commutator Let$I$be the$2\times 2$identity matrix. Then prove that$-I$cannot be a commutator$[A, B]:=ABA^{-1}B^{-1}$for any$2\times 2$matrices$A$and$B$with determinant$1$. Proof. Assume that$[A, B]=-I$. Then$ABA^{-1}B^{-1}=-I$implies $ABA^{-1}=-B. […] • Find Inverse Matrices Using Adjoint Matrices Let A be an n\times n matrix. The (i, j) cofactor C_{ij} of A is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}),$ where$M_{ij}$is the$(i,j)$minor matrix obtained from$A$removing the$i$-th row and$j$-th column. Then consider the$n\times n\$ matrix […]