Block diagonalization and Jordan blocks

Which of the following is FALSE?

• All matrices are block diagonalizable.

• All matrices are unitarily block diagonalizable.

• All matrices are unitarily upper triangularizable.

• All matrices are similar to a block diagonal matrix with unispectral blocks.

Which of the following is FALSE?

• rank $A\ge$ rank $A^2\ge$ rank $A^3\ge\cdots$

• image $A\supseteq$ image $A^2\supseteq$ image $A^3\supseteq\cdots$

• If rank $A^3=$ rank $A^5$ , then both are equal to rank $A^6$

• If rank $A^3>$ rank $A^4$ , then rank $A^2<$ rank $A$

• If rank $I>$ rank $A$ , then $A$ is nilpotent.

Which of the following is FALSE?

• A matrix is diagonal iff $A=J_1\left(\lambda_1\right)\oplus J_1\left(\lambda_2\right)\oplus\cdots\oplus J_1\left(\lambda_n\right)$

• A matrix is diagonalizable iff its Jordan canonical form is $J_1\left(\lambda_1\right)\oplus J_1\left(\lambda_2\right)\oplus\cdots\oplus J_1\left(\lambda_n\right)$

• $J_k\left(\lambda\right)$ is not diagonalizable for k>1

• Two matrices are similar iff they have the same Jordan canonical form up to permuting factors.

• $J_2\left(\lambda\right)\oplus J_3\left(\mu\right)$ is not similar to $J_3\left(\mu\right)\oplus J_2\left(\lambda\right)$

Which of the following is FALSE?

• The proof of the block diagonalization theorem used Sylvester's linear matrix equation theorem.

• It is because of the block diagonalization theorem that we can focus on unispectral matrices when we construct the Jordan canonical form.

• We can use the sequence of rank inequalities to detect nilpotentness.

• Any unispectral matrix can be made nilpotent by subtracting a multiple of the identity matrix.

• The Jordan canonical form of a matrix is unique up to permuting the factors.