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Linear Algebra

A Matrix Equation of a Symmetric Matrix and the Limit of its Solution

Problem 457

Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.

(a) Prove that for sufficiently small positive real $\epsilon$, the equation
\[A\mathbf{x}+\epsilon\mathbf{x}=\mathbf{v}\] has a unique solution $\mathbf{x}=\mathbf{x}(\epsilon) \in \R^n$.

(b) Evaluate
\[\lim_{\epsilon \to 0^+} \epsilon \mathbf{x}(\epsilon)\] in terms of $\mathbf{v}$, the eigenvectors of $A$, and the inner product $\langle\, ,\,\rangle$ on $\R^n$.

 
(University of California, Berkeley, Linear Algebra Qualifying Exam)

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Linear Algebra

Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like.

Problem 381

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 $n$ large enough so that the length $\|A^n\mathbf{v}\|$ is as small as we like.

(University of California, Berkeley, Linear Algebra Final Exam Problem)
 
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Linear Algebra

If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set?

Problem 317

Suppose that $A$ is a real $n\times n$ matrix.

(a) Is it true that $A$ must commute with its transpose?

(b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set.
Is it true that the rows of $A$ must also form an orthonormal set?

(University of California, Berkeley, Linear Algebra Qualifying Exam)

 
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Linear Algebra

Find the Rank of the Matrix $A+I$ if Eigenvalues of $A$ are $1, 2, 3, 4, 5$

Problem 35

Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix.

(UCB-University of California, Berkeley, Exam)

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Linear Algebra

Linear Dependent/Independent Vectors of Polynomials

Problem 15

Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent?

(a) At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ for $i=1,2,3,4$.

(b) At $0$ each of the polynomials has the value $1$. Namely $p_i(0)=1$ for $i=1,2,3,4$.

(University of California, Berkeley)

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