Spectral Theorem

Consider the vectors in $\mathbb {C}^{2}$ (sometimes called spinors)

Given a $4 \times 4$ symmetric matrix $A$ over $\mathbb {R}$ with the eigenvalues $\lambda_{1} = 0$, $\lambda_{2} = 1$, $\lambda_{3} = 2$, $\lambda_{4} = 3$ and the corresponding normalized eigenvectors

Reconstruct the matrix $A$ using the spectral theorem. The eigenvectors given above are called Bell basis.

Let $z \in \mathbb {C}$ and $A$ be an $n \times n$ normal matrix with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ and pairwise orthonormal eigenvectors $\overrightarrow {v}_{1}, \ldots , \overrightarrow {v}_{n}$. Use the spectral decomposition to calculate $\mathrm{exp}\left(zA\right)$.

Let $A$ be a hermitian $n \times n$ matrix. Assume that all the eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ are pairwise different. Then the normalized eigenvectors $\overrightarrow {u}_{j}$ ($j = 1, \ldots , n$) satisfy $\overrightarrow {u}_{j}^{*}\overrightarrow {u}_{k} = 0$ for $j \neq k$ and $\overrightarrow {u}_{j}^{*}\overrightarrow {u}_{j} = 1$. We have

Let $\overrightarrow {e}_{k}$ ($k = 1, \ldots , n$) be the standard basis in $\mathbb {C}^{n}$. Calculate $U^{*}AU$, where $U = \sum_{k=1}^{n} \overrightarrow {u}_{k}\overrightarrow {e}_{k}^{*}$.

Let $A$ be a positive definite $n \times n$ matrix. Thus all the eigenvalues are real and positive. Assume that all the eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ are pairwise different. Then the normalized eigenvectors $\overrightarrow {u}_{j}$ ($j = 1, \ldots , n$) satisfy $\overrightarrow {u}_{j}^{*}\overrightarrow {u}_{k} = 0$ for $j \neq k$ and $\overrightarrow {u}_{j}^{*}\overrightarrow {u}_{j} = 1$. We have (spectral theorem)

Let $\overrightarrow {e}_{k}$ ($k = 1, \ldots , n$) be the standard basis in $\mathbb {C}^{n}$. Calculate $\ln (A)$. Note that the unitary matrix

transforms $A$ into a diagonal matrix, i.e. $\widetilde{A} = U^{*}AU$ is a diagonal matrix.

Consider the normal noninvertible $3 \times 3$ matrix

Consider the two $3 \times 3$ permutation matrices with $P_{1} = P_{2}^{T}$

Consider the unitary matrix

Find the skew-hermitian matrix $K$ such that $U = \mathrm{exp}\left(K\right)$. Utilize the spectral representation.

Consider the Hadamard matrix

with eigenvalues $+1$ and $-1$. Find the square roots of this matrix applying the spectral theorem. Since $\sqrt{1} = \pm 1$ and $\sqrt{-1} = \pm i$ four cases have to be considered

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$. An $n \times n$ matrix $B$ is called a square root of $A$ if $B^{2} = A$. Find the square roots of the $2 \times 2$ identity matrix applying the spectral theorem. The eigenvalues of $I_{2}$ are $\lambda_{1} = 1$ and $\lambda_{2} = 1$. As normalized eigenvectors choose

which form an orthonormal basis in $\mathbb {C}^{2}$. Four cases

have to be studied. The first and last cases are trivial. So study the second case $(\sqrt{\lambda_{1}}, \sqrt{\lambda_{2}}) = (1, -1)$. The second case and the third case are ``equivalent''.

Let $\lambda_{1}, \lambda_{2}, \mu_{1}, \mu_{2} \in \mathbb {C}$. Let $\overrightarrow {v}_{1}$, $\overrightarrow {v}_{2}$ (column vectors) be an orthonormal basis in $\mathbb {C}^{2}$. We define the $2 \times 2$ matrices (spectral theorem)

Find the commutator $[A, B]$. Find the conditions on $\lambda_{1}$, $\lambda_{2}$, $\mu_{1}$, $\mu_{2}$ such that $[A, B] = 0_{2}$.

Let $\overrightarrow {v}_{0}, \overrightarrow {v}_{1}, \ldots , \overrightarrow {v}_{2^{n}-1}$ be an orthonormal basis in $\mathbb {C}^{2^{n}}$. We define

Show that $U$ is unitary. In other words show that $UU^{*} = I_{2^{n}}$, using the completeness relation

The star product of the Hadamard matrix $U_{H}$ with itself provides the Bell matrix

which is a unitary matrix. Note that the eigenvalues of the Bell matrix are $+1$ (twice) and $-1$ (twice).

Let $A$ be a normal matrix with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ and pairwise orthonormal eigenvectors $\overrightarrow {a}_{j}$ (column vectors). Let $B$ be a normal matrix with eigenvalues $\mu_{1}, \ldots , \mu_{n}$ and pairwise orthonormal eigenvectors $\overrightarrow {b}_{j}$ (column vectors). Then $A$ and $B$ can be written as

Let $z \in \mathbb {C}$. Use the spectral decomposition to calculate $e^{zA}Be^{-zA}$.

Consider the normalized vectors

in $\mathbb {R}^{3}$. Find the eigenvalues and eigenvectors of the $3 \times 3$ matrix $M = 1 \cdot \overrightarrow {v}_{1}\overrightarrow {v}_{1}^{*} + 0 \cdot \overrightarrow {v}_{2}\overrightarrow {v}_{2}^{*} - 1 \cdot \overrightarrow {v}_{3}\overrightarrow {v}_{3}^{*}$ without calculating $M$.

Let $A$ be an $n \times n$ normal matrix with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ and pairwise normalized orthogonal eigenvectors $\overrightarrow {v}_{j}$ ($j = 1, \ldots , n$). Then

Show that $\cos (A) = \sum_{j=1}^{n} \cos (\lambda_{j})\overrightarrow {v}_{j}\overrightarrow {v}_{j}^{*}$, $\sin (A) = \sum_{j=1}^{n} \sin (\lambda_{j})\overrightarrow {v}_{j}\overrightarrow {v}_{j}^{*}$.

The spectral theorem for $n \times n$ normal matrices over $\mathbb {C}$ is as follows: A matrix $A$ is normal if and only if there exists an $n \times n$ unitary matrix $U$ and a diagonal matrix $D$ such that $D = U^{*}AU$. Use this theorem to prove that the matrix

is nonnormal.

Find all $2 \times 2$ matrices over $\mathbb {C}$ which admit the normalized eigenvectors

with the corresponding eigenvalues $\lambda_{1}$ and $\lambda_{2}$.

Consider the nonnormal $2 \times 2$ matrix

with eigenvalues $\lambda_{1} = +1$ and $\lambda_{2} = -1$ and normalized eigenvectors

The eigenvectors are linearly independent, but not orthonormal. Calculate $\overrightarrow {v}_{1}^{*}\overrightarrow {v}_{2}$ and

Find $\left\| A - B\right\|$. Discuss. Is the matrix $B$ normal?