§1.3 Determinants, Linear Operators, and Spectral Expansions

The notation is that of (1.2.58). For $n=2$:

For $n=3$:

Higher-order determinants are natural generalizations. The minor $M_{jk}$ of the entry $a_{jk}$ in the $n$th-order determinant $det[a_{jk}]$ is the ($n-1$)th-order determinant derived from $det[a_{jk}]$ by deleting the $j$th row and the $k$th column. The cofactor $A_{jk}$ of $a_{jk}$ is

An $n$th-order determinant expanded by its $j$th row is given by

If two rows (or columns) of a determinant are interchanged, then the determinant changes sign. If two rows (columns) of a determinant are identical, then the determinant is zero. If all the elements of a row (column) of a determinant are multiplied by an arbitrary factor $\mu$, then the result is a determinant which is $\mu$ times the original. If $\mu$ times a row (column) of a determinant is added to another row (column), then the value of the determinant is unchanged.

The determinant of an upper or lower triangular, or diagonal, square matrix $𝐀$ is the product of the diagonal elements $det(𝐀)={\prod }_{i=1}^n{a}_{ii}$.

For real-valued $a_{jk}$,

Compare also (1.3.7) for the left-hand side. Equality holds iff

for every distinct pair of $j,k$, or when one of the factors ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes.

where ${\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n$ are the $n$th roots of unity (1.11.21).

For

Square matices can be seen as linear operators because $𝐀(\alpha 𝐚+\beta 𝐛)=\alpha 𝐀𝐚+\beta 𝐀𝐛$ for all $\alpha ,\beta \in ℂ$ and $𝐚,𝐛\in {𝐄}_n$, the space of all $n$-dimensional vectors.

The adjoint of a matrix $𝐀$ is the matrix ${𝐀}^{\ast }$ such that $⟨𝐀𝐚,𝐛⟩=⟨𝐚,{𝐀}^{\ast }𝐛⟩$ for all $𝐚,𝐛\in {𝐄}_n$. In the case of a real matrix ${𝐀}^{\ast }={𝐀}^{\mathrm{T}}$ and in the complex case ${𝐀}^{\ast }={𝐀}^{\mathrm{H}}$.

Real symmetric ($𝐀={𝐀}^{\mathrm{T}}$) and Hermitian ($𝐀={𝐀}^{\mathrm{H}}$) matrices are self-adjoint operators on ${𝐄}_n$. The spectrum of such self-adjoint operators consists of their eigenvalues, ${\lambda }_i,i=1,2,\mathrm{\dots },n$, and all ${\lambda }_i\in ℝ$. The corresponding eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$ can be chosen such that they form a complete orthonormal basis in ${𝐄}_n$.

Let the columns of matrix $𝐒$ be these eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$, then ${𝐒}^{-1}={𝐒}^{\mathrm{H}}$, and the similarity transformation (1.2.73) is now of the form ${𝐒}^{\mathrm{H}}𝐀𝐒={\lambda }_i{\delta }_{i,j}$. For Hermitian matrices $𝐒$ is unitary, and for real symmetric matrices $𝐒$ is an orthogonal transformation.

For self-adjoint $𝐀$ and $𝐁$, if $[𝐀,𝐁]=𝟎$, see (1.2.66), simultaneous eigenvectors of $𝐀$ and $𝐁$ always exist.

Assuming $\{{𝐚}_i\}$ is an orthonormal basis in ${𝐄}_n$, any vector $𝐮$ may be expanded as

Taking $l^2$ norms,

which is Parseval’s equality.