Question

In each case find the characteristic polynomial, eigenvalues, eigenvectors, and (if possible) an invertible matrix \(P\) such that \(P^{-1} A P\) is diagonal. a. \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 2\end{array}\right]\) b. \(A=\left[\begin{array}{rr}2 & -4 \\ -1 & -1\end{array}\right]\) c. \(A=\left[\begin{array}{rrr}7 & 0 & -4 \\ 0 & 5 & 0 \\ 5 & 0 & -2\end{array}\right]\) d. \(A=\left[\begin{array}{rrr}1 & 1 & -3 \\ 2 & 0 & 6 \\ 1 & -1 & 5\end{array}\right]\) e. \(A=\left[\begin{array}{rrr}1 & -2 & 3 \\ 2 & 6 & -6 \\ 1 & 2 & -1\end{array}\right]\) f. \(A=\left[\begin{array}{lll}0 & 1 & 0 \\ 3 & 0 & 1 \\ 2 & 0 & 0\end{array}\right]\) g. \(A=\left[\begin{array}{rrr}3 & 1 & 1 \\ -4 & -2 & -5 \\ 2 & 2 & 5\end{array}\right]\) h. \(A=\left[\begin{array}{rrr}2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & -1 & 2\end{array}\right]\) i. \(A=\left[\begin{array}{lll}\lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \mu\end{array}\right], \lambda \neq \mu\)

Short answer

Compute characteristic polynomial, find eigenvalues and eigenvectors. If possible, use them to form an invertible matrix \( P \) such that \( P^{-1}AP \) is diagonal.

Step-by-step solution

Compute Characteristic Polynomial

To find the characteristic polynomial of a matrix \( A \), start by setting up the equation \( \det(A - \lambda I) = 0 \), where \( \lambda \) is a scalar and \( I \) is the identity matrix of the same dimension as \( A \). Calculate the determinant and simplify to get a polynomial in terms of \( \lambda \).

Solve for Eigenvalues

The eigenvalues of matrix \( A \) are the solutions to the characteristic polynomial found in Step 1. Solve the polynomial equation for \( \lambda \) to find the eigenvalues.

Find Eigenvectors for Each Eigenvalue

Once an eigenvalue \( \lambda_i \) is identified, substitute it into the equation \( (A - \lambda_i I)x = 0 \). Solve the resulting system of linear equations to find the eigenvector(s) \( x \) corresponding to \( \lambda_i \). Repeat for each eigenvalue.

Check Diagonalizability and Find Matrix P

A matrix \( A \) is diagonalizable if there are enough linearly independent eigenvectors to form a basis for the space. If \( A \) is \( n \times n \), find \( n \) linearly independent eigenvectors. If possible, construct the matrix \( P \) whose columns are these eigenvectors. Diagonal matrix \( D \) is formed with the eigenvalues on the diagonal, and the relation \( P^{-1}AP = D \) holds.

Key concepts

Characteristic Polynomial

The characteristic polynomial is a central concept when dealing with matrices in linear algebra. To find it, you need to start with the equation \( \det(A - \lambda I) = 0 \). Here, \( A \) is your matrix, \( \lambda \) is a scalar, and \( I \) is the identity matrix of the same size as \( A \). By solving this equation, you convert your matrix into a polynomial in terms of \( \lambda \). The determinant calculation involves subtraction of \( \lambda I \) from \( A \), yielding a matrix whose determinant can be calculated through expansion along a row or column, or using more advanced determinant tricks like cofactor expansion or row operations.

This polynomial is essential because its roots will be the eigenvalues of the matrix. The degree of the characteristic polynomial corresponds to the size of the matrix; thus, a 2x2 matrix will have a quadratic characteristic polynomial, whereas a 3x3 matrix will have a cubic one.

Diagonalization

Diagonalization is the process of transforming a matrix into a diagonal form. A matrix is diagonalizable if and only if it can be written in the form \( P^{-1}AP = D \), where \( D \) is a diagonal matrix and \( P \) is the matrix formed by the eigenvectors of \( A \). This transformation is particularly useful because diagonal matrices are much easier to work with, especially in powers and exponentials of matrices.

Not all matrices are diagonalizable. For a matrix to be diagonalizable, it needs enough linearly independent eigenvectors to form a complete basis for the space. Specifically, for an \( n \times n \) matrix \( A \), there must be \( n \) linearly independent eigenvectors. If a matrix does not meet this criterion, it cannot be diagonalized over the real numbers, though it might be possible over the complex numbers.

Matrix Algebra

Matrix algebra is the set of algebraic rules and operations associated with matrices. Operations include matrix addition, subtraction, multiplication, and finding determinants and inverses. Each operation follows specific properties, such as associativity and distributivity, which parallel those of number algebra but with some differences (e.g., matrix multiplication is not commutative).

• **Addition/Subtraction**: These operations are done element-wise between matrices of the same size.


• **Multiplication**: It involves the dot product between rows and columns of two matrices and requires that the number of columns of the first matrix matches the number of rows of the second.


• **Determinant**: A scalar value that can determine if a matrix has an inverse and is used in the characteristic polynomial.


• **Inverse**: Exists only if a matrix is square and non-singular (has a non-zero determinant). The inverse satisfies \( A^{-1}A = I \).

Understanding these operations is fundamental to solving problems involving eigenvalues and eigenvectors, as they leverage these rules to manipulate and factor matrices.

Linear Independence

Linear independence is a concept that indicates whether a set of vectors is independent or dependent. A set of vectors is said to be linearly independent if none of the vectors can be written as a linear combination of the others. In contrast, they are linearly dependent if at least one vector in the set can be expressed like this.

To determine the linear independence of a set of vectors, arrange them as columns in a matrix and row reduce the matrix to its reduced row-echelon form (RREF). If every column contains a leading 1 (also known as a pivot), the vectors are linearly independent.

In terms of matrices and diagonalization, finding enough linearly independent eigenvectors is crucial. These eigenvectors form the columns of matrix \( P \) in the transformation \( P^{-1}AP = D \). If you can find \( n \) linearly independent eigenvectors for an \( n \times n \) matrix, it confirms that the matrix is diagonalizable. This makes understanding linear independence essential for solving many linear algebra problems.