Let $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ are vectors in $\R^n$. Suppose that vectors $\mathbf{u}_1$, $\mathbf{u}_2$ are orthogonal and the norm of $\mathbf{u}_2$ is $4$ and $\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number $a$ in $\mathbf{u_1}=\mathbf{u_2}+a\mathbf{u}_3$.
(The Ohio State University, Linear Algebra Exam Problem)
The inner product (dot product) of two vectors $\mathbf{v}_1, \mathbf{v}_2$ is defined to be
\[\mathbf{v}_1\cdot \mathbf{v}_2 :=\mathbf{v}^{\trans}_1\mathbf{v}_2.\]
Two vectors $\mathbf{v}_1, \mathbf{v}_2$ are orthogonal if the inner product
\[\mathbf{v}_1\cdot \mathbf{v}_2=0.\]
The norm (length, magnitude) of a vector $\mathbf{v}$ is defined to be
\[||\mathbf{v}||=\sqrt{\mathbf{v}\cdot \mathbf{v}}.\]
Solution.
We first express the given conditions in term of inner products (dot products).
Since $\mathbf{u}_1$ and $\mathbf{u}_2$ are orthogonal, the inner product
\[\mathbf{u}_2\cdot \mathbf{u}_1=\mathbf{u}_1 \cdot \mathbf{u}_2=0. \tag{a}\]
Also, since the norm of $\mathbf{u}_2$ is $4$, we obtain
\[\mathbf{u}_2\cdot\mathbf{u}_2=||\mathbf{u}_2||^2=16. \tag{b}\]
The last condition can be written as
\[\mathbf{u}_2\cdot \mathbf{u}_3=\mathbf{u}_2^{\trans}\mathbf{u}_3=7 \tag{c}.\]
Now we compute the inner product $\mathbf{u}_2$ and $\mathbf{u}_1$.
\begin{align*}
0\stackrel{(a)}{=}&\mathbf{u}_2\cdot \mathbf{u}_1 =\mathbf{u}_2\cdot (\mathbf{u}_2+a\mathbf{u}_3)\\
&=\mathbf{u}_2\cdot \mathbf{u}_2+a\mathbf{u}_2\cdot \mathbf{u}_3\\
&\stackrel{(b), (c)}{=}16+7a.
\end{align*}
Therefore, solving this we obtain
\[a=-\frac{16}{7}.\]
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.
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\[ \mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} , \mathbf{v}_2 = \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} , \mathbf{v}_3 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} . \]
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Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\R^n$ such that their length are
\[\|\mathbf{a}\|=\|\mathbf{b}\|=1\]
and the inner product
\[\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.\]
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Show that eigenvalues of a Hermitian matrix $A$ are real numbers.
(The Ohio State University Linear Algebra Exam Problem)
We give two proofs. These two proofs are essentially the same.
The second proof is a bit simpler and concise compared to the first one.
[…]
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(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.
Prove that $P$ is an idempotent matrix.
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Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if
\[A_1^2+A_2^2+\cdots+A_m^2=\calO,\]
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(c) Find an orthonormal basis of the row space of $A$.
(The Ohio State University, Linear Algebra Exam […]