Diagonalize the following matrices: (a) ( 3 -9 2 -6 ), (b) ( 5 -4

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Diagonalize the following matrices: (a) ( 3 -9 ...

Diagonalize the following matrices: (a) $\left(\begin{array}{ll}3 & -9 \\ 2 & -6\end{array}\right)$, (b) $\left(\begin{array}{ll}5 & -4 \\ 2 & -1\end{array}\right)$, (c) $\left(\begin{array}{rr}-4 & -2 \\ 5 & 2\end{array}\right)$, (d) $\left(\begin{array}{rrr}-2 & 3 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 3\end{array}\right)$. (e) $\left(\begin{array}{rrr}8 & 0 & -3 \\ -3 & 0 & -1 \\ 3 & 0 & -2\end{array}\right)$, (f) $\left(\begin{array}{rrr}3 & 3 & 5 \\ 5 & 6 & 5 \\ -5 & -8 & -7\end{array}\right)$, (g) $\left(\begin{array}{rrr}2 & 5 & 5 \\ 0 & 2 & 0 \\ 0 & -5 & -3\end{array}\right)$, (h) $\left(\begin{array}{rrrr}1 & 0 & -1 & 1 \\ 0 & 2 & -1 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -2\end{array}\right)$, (i) $\left(\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right)$, (j) $\left(\begin{array}{rrrr}2 & 1 & -1 & 0 \\ -3 & -2 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & -1\end{array}\right)$.

Diagonalize the following matrices:
(a) $\left(\begin{array}{ll}3 & -9 \\ 2 & -6\end{array}\right)$,
(b) $\left(\begin{array}{ll}5 & -4 \\ 2 & -1\end{array}\right)$,
(c) $\left(\begin{array}{rr}-4 & -2 \\ 5 & 2\end{array}\right)$,
(d) $\left(\begin{array}{rrr}-2 & 3 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 3\end{array}\right)$.
(e) $\left(\begin{array}{rrr}8 & 0 & -3 \\ -3 & 0 & -1 \\ 3 & 0 & -2\end{array}\right)$,
(f) $\left(\begin{array}{rrr}3 & 3 & 5 \\ 5 & 6 & 5 \\ -5 & -8 & -7\end{array}\right)$,
(g) $\left(\begin{array}{rrr}2 & 5 & 5 \\ 0 & 2 & 0 \\ 0 & -5 & -3\end{array}\right)$,
(h) $\left(\begin{array}{rrrr}1 & 0 & -1 & 1 \\ 0 & 2 & -1 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -2\end{array}\right)$,
(i) $\left(\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right)$,
(j) $\left(\begin{array}{rrrr}2 & 1 & -1 & 0 \\ -3 & -2 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & -1\end{array}\right)$.

Key Concepts

Eigenvalues

An eigenvalue is a scalar ? such that for a given square matrix A, there exists a nonzero vector v (called the eigenvector) satisfying the equation Av = ?v. In the process of diagonalization, determining the eigenvalues by solving the characteristic equation det(A - ?I) = 0 is the crucial first step, as they form the diagonal entries of the resulting diagonal matrix.

Eigenvectors

An eigenvector associated with a specific eigenvalue ? is a nonzero vector that satisfies Av = ?v. Once the eigenvalues are found, the corresponding eigenvectors are computed by solving (A - ?I)v = 0 for each eigenvalue. These eigenvectors are used to form the columns of the transformation matrix used in the diagonalization process.

Characteristic Polynomial

The characteristic polynomial of a matrix A is obtained from the determinant of (A - ?I) and is set equal to zero in order to find the eigenvalues. This polynomial encapsulates key information about the matrix, such as its eigenvalues and their algebraic multiplicities, and hence plays a central role in the diagonalization process.

Algebraic and Geometric Multiplicities

Algebraic multiplicity refers to the number of times an eigenvalue appears as a root of the characteristic polynomial, while geometric multiplicity is the dimension of the eigenspace corresponding to that eigenvalue (i.e., the number of linearly independent eigenvectors). A matrix is diagonalizable if, for each eigenvalue, the geometric multiplicity equals the algebraic multiplicity, ensuring there are enough linearly independent eigenvectors to form a basis.

Similarity Transformation

A similarity transformation involves writing a matrix A as PDP?¹, where D is a diagonal matrix and P is an invertible matrix whose columns are the eigenvectors of A. This transformation shows that A and D are similar matrices and share the same eigenvalues. Diagonalization is essentially the process of finding such a similarity transformation.

Diagonalization Process

Diagonalization is the process of finding a diagonal matrix D and an invertible matrix P such that A = PDP?¹. It involves computing the eigenvalues and eigenvectors of A, constructing the matrix P with eigenvectors as columns, and then forming D with the corresponding eigenvalues along its diagonal. This process simplifies many matrix computations, such as raising A to a power, since powers of D are straightforward to calculate.

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