# Princeton-university-eye-catch

by Yu ·

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- Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.) Solution. For example, let $A$ be the following $3\times […]
- 7 Problems on Skew-Symmetric Matrices Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$. (a) Prove that $A+B$ is skew-symmetric. (b) Prove that $cA$ is skew-symmetric for any scalar $c$. (c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is […]
- Given a Spanning Set of the Null Space of a Matrix, Find the Rank Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors \[\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}, \text{ and } \begin{bmatrix} 1 \\ -1 \\ 0 […]
- Orthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct Eigenvalues Suppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$. Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$. (Nagoya University, Linear Algebra Final Exam Problem) Hint. Two […]
- The Sum of Cosine Squared in an Inner Product Space Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$. Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$. Prove that \[\cos […]
- Determine All Matrices Satisfying Some Conditions on Eigenvalues and Eigenvectors Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors \[\begin{bmatrix} 1 \\ 0 \end{bmatrix} \text{ and } \begin{bmatrix} 2 \\ 1 \end{bmatrix},\] respectively. Solution. Suppose […]
- Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less. Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{align*} (a) […]
- If Two Matrices Have the Same Eigenvalues with Linearly Independent Eigenvectors, then They Are Equal Let $A$ and $B$ be $n\times n$ matrices. Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$. Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly […]