Spectral Methods
Spectral methods refer to the study of matrices.
Identity Matrix
The identity matrix $I$ is a matrix which is all zero apart from the lead diagonal:
\[\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}\]For any matrix $A$:
\[A\times I = I \times A = A\]This is valid for square matrices.
Matrix Inverse
For an $n\times n$ matrix $A$ its inverse is the matrix denoted by $A^{-1}$ for which:
\[A\times A^{-1}=A^{-1}\times A=I\]Not every matrix has an inverse.
- Singular matrices have an inverse.
- Non-singular matrices don’t have an inverse.
Matrix Determinant
\[\det A \neq 0 \iff A^{-1} \text{ exists}\]This is to say that $A$ is non-singular.
Eigenvalues
Given an $n\times n$ matrix $A$, find all scalr values $\lambda$ for which ther is an $n$-vector $\vec x_\lambda$ satisfying:
\[A\times \vec x_\lambda = \lambda \vec x_\lambda\]- These scalar values are called the eigenvalues of $A$.
- The $n$-vectors $\vec x_\lambda$ are called the eigenvectors of $A$ (with respect to $\lambda$).
The case $\vec x = \vec 0$ always produces a solution. Theretofore, in finding eigenvectors, this case is not considered.
The case $\lambda = 0$ is of interest.
In general, finding eigenvalues involves fiding $\lambda$ for which:
\[(A-\lambda I)\vec x_\lambda = \vec 0\]This is the same as finding:
\[\det(A-\lambda I)=0\]Notice that $(A-\lambda I)$ is the matrix $A$ with $\lambda$ subtracted from each diagonal entry.
Solving via Polynomial Roots
The form $\det(A-\lambda I)=0$ has an interpretation involving roots of a polynomial.
Since $\det A$ (with some parameter $\lambda$) may be described as a polynomial of degree $n$ in $\lambda$ (with $A$ an $n\times n$ matrix) the solutions to $\det(A-\lambda I)=0$ are exactly the roots of this polynomial which is called the characteristic polynomial of $A$.
The characteristic polynomial of $A$ is denoted by: $\chi(A)$.
Consequences of Polynomial Similarity
- An $n\times n$ matrix has $n$ eigenvalues (although these are not necessarily distinct).
- This is similar to any other polynomial.
- Eigenvalues may be complex numbers.
- We won’t be studying these.
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Matrices with all real eigenvalues include symmetric matrices:
\[[a_{ij}]=[a_{ji}]\]
Ordering Eigenvalues and Dominance
Conventionally eigenvalues are written as:
\[\sigma(A)=(\lambda_1,\lambda_2,\ldots,\lambda_k,\ldots,\lambda_n)\]This is called the spectrum of $A$.
- The eigenvalues are ordered from largest to smallest.
- For complex values, the modulus is used.
- If $\lambda_1$ is unique, then it is called the dominant eigenvalue.
- The largest eigenvalue is referred to as the spectral radius.
- For complex values if $\lambda$ is an eigenvalue of $A$ then the conjugate, $\bar\lambda$, is too.