A, B, P, and D are n ×n matrices. Mark each statement True...

$A, B, P,$ and $D$ are $n \times n$ matrices. Mark each statement True or False. Justify each answer. (Study Theorems 5 and 6 and the examples in this section carefully before you try these exercises.) a. $A$ is diagonalizable if $A$ has $n$ eigenvectors. b. If $A$ is diagonalizable, then $A$ has $n$ distinct eigenvalues. c. If $A P=P D,$ with $D$ diagonal, then the nonzero columns of $P$ must be eigenvectors of $A .$ d. If $A$ is invertible, then $A$ is diagonalizable.

$A, B, P,$ and $D$ are $n \times n$ matrices. Mark each statement True or False. Justify each answer. (Study Theorems 5 and 6 and the examples in this section carefully before you try these exercises.)
a. $A$ is diagonalizable if $A$ has $n$ eigenvectors.
b. If $A$ is diagonalizable, then $A$ has $n$ distinct eigenvalues.
c. If $A P=P D,$ with $D$ diagonal, then the nonzero columns of $P$ must be eigenvectors of $A .$
d. If $A$ is invertible, then $A$ is diagonalizable.

Key Concepts

Similarity Transformation

A similarity transformation is an expression of the form A = PDP?¹, where P is an invertible matrix whose columns are eigenvectors of A and D is a diagonal matrix containing the corresponding eigenvalues. This concept shows that matrices can be 'similar' even if they are not identical, and that diagonalizability is linked to finding the right basis of eigenvectors.

Eigenvalue Multiplicity

Eigenvalue multiplicity comes in two forms: algebraic multiplicity, which is the number of times an eigenvalue appears as a root of the characteristic polynomial, and geometric multiplicity, which is the number of linearly independent eigenvectors corresponding to an eigenvalue. A matrix is diagonalizable if, for every eigenvalue, the geometric multiplicity equals the algebraic multiplicity, even if the eigenvalues are not distinct.

Invertible Matrices and Diagonalizability

While invertibility of a matrix assures that the matrix has a nonzero determinant (and consequently, no zero eigenvalues), it does not imply that the matrix is diagonalizable. Diagonalizability depends on whether there exists a complete set of linearly independent eigenvectors, regardless of the invertibility status. Thus, an invertible matrix may still fail to be diagonalizable if it does not have a basis of eigenvectors.

Matrix Diagonalization

Matrix diagonalization refers to the process of finding an invertible matrix P and a diagonal matrix D such that A = PDP?¹. A matrix is diagonalizable if it has a full set of linearly independent eigenvectors, meaning that the dimensions of the eigenspaces add up to n. Diagonalization simplifies many problems in linear algebra, such as computing powers of matrices or solving differential equations.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts where an eigenvector of a matrix A is a nonzero vector that, when multiplied by A, results in a scalar multiple of itself; that scalar is called the eigenvalue. They are critical in determining the structure of a matrix and play a key role in diagonalization, as a matrix that is diagonalizable must have enough eigenvectors to form a basis.

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