# idempotent-matrix

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• Sequence Converges to the Largest Eigenvalue of a Matrix Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$. Furthermore, suppose that $|\lambda_1| > |\lambda_2| \geq \cdots \geq […] • Orthonormal Basis of Null Space and Row Space Let A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}. (a) Find an orthonormal basis of the null space of A. (b) Find the rank of A. (c) Find an orthonormal basis of the row space of A. (The Ohio State University, Linear Algebra Exam […] • Determine Whether Given Matrices are Similar (a) Is the matrix A=\begin{bmatrix} 1 & 2\\ 0& 3 \end{bmatrix} similar to the matrix B=\begin{bmatrix} 3 & 0\\ 1& 2 \end{bmatrix}? (b) Is the matrix A=\begin{bmatrix} 0 & 1\\ 5& 3 \end{bmatrix} similar to the matrix […] • Basis of Span in Vector Space of Polynomials of Degree 2 or Less Let P_2 be the vector space of all polynomials of degree 2 or less with real coefficients. Let \[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}$ be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$. (Linear […]
• If $A$ is an Idempotent Matrix, then When $I-kA$ is an Idempotent Matrix? A square matrix $A$ is called idempotent if $A^2=A$. (a) Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be the $n\times n$ identity matrix. Prove that the matrix $I-A$ is an idempotent matrix. (b) Assume that $A$ is an $n\times n$ nonzero idempotent matrix. Then […]
• The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$. Then find the matrix representation of the linear transformation $T$ with respect to the […]
• Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like. Consider the matrix $A=\begin{bmatrix} 3/2 & 2\\ -1& -3/2 \end{bmatrix} \in M_{2\times 2}(\R).$ (a) Find the eigenvalues and corresponding eigenvectors of $A$. (b) Show that for $\mathbf{v}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}\in \R^2$, we can choose […]
• Is the Map $T (f) (x) = f(x) – x – 1$ a Linear Transformation between Vector Spaces of Polynomials? Let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_4 \rightarrow \mathrm{P}_{4}$ be the map defined by, for $f \in \mathrm{P}_4$, \[ […]