Eigenvalues and eigenvectors linear algebra conceptual questions

eigenvalues-eigenvectorslinear algebramatrices

  1. If $A$ is a square matrix and $P$ is an invertible matrix each of whose columns is an eigenvector of $A$, then: $P^{-1}AP$ is a diagonal matrix. True or false.

I am not sure of this question but I think that it must be could be false because if $P^{-1}AP = D$, then $AP = PD$, and wouldn't these yield different results?

  1. Let $T: V \to V$ be a linear transformation and let $B$ be a basis for $V$. Then $T$ and $[T]B$ have the same eigenvalues, but may have different vectors. True or False

I believe this answer is true because $T$ relative to the basis $B$ for $V$ shouldn't have different eigenvalues than $T$ right?

  1. A square matrix $A$ is diagonalizable if and only if for each eigenvalue $c$ of $A$ the algebraic multiplicity of $c$ is equal to the dimension of the eigenspace of $A$ corresponding to $c$. True or False.

I believe this answer to be true because if the multiplicity of $c$ is one, then there would be one eigenvector for $c$, and therefore the eigenspace of $A$ would have the same dimension as $c$ which is one correct?

  1. The dimension of the eigenspace of a matrix $A$ corresponding to an eigenvalue $c$ is equal to the rank of $A – cI$. True or false

This one I am stumped on, I believe it is true but I am not sure

  1. true or false. A square matrix $A$ is singular if and only if $0$ is an eigenvalue of $A$.

I know that this is true if there is a determinant of $0$, but is it also for an eigenvalue? I believe it is but wanted to check.

Please answer any or all of these questions, any help is very appreciated.

Best Answer

Question 1: The statement is true. As you said, $P^{-1}AP = D$ is equivalent to $$ AP = PD. $$ If $D$ is diagonal, then what is the relationship between the columns of $P$ and the columns of $PD$? What is the relationship between the columns of $P$ and the columns of $AP$?

Question 2: Indeed, the statement is true.

Question 3: Indeed, the statement is true. However, your justification for the statement doesn't make any sense.

Question 4: The statement is false. The eigenspace is the nullspace of $A - cI$, but the rank of $A - cI$ is the dimension of the column space of $A - cI$. For example, with $c = 1$ and the diagonal matrix $$ A = \pmatrix{1\\&2\\&&3}, $$ we find that the rank of $A - cI$ is $2$, but the dimension of the associated eigenspace is $1$.

Question 5: Indeed, the statement is true.

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