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# Tag Archives: Linear Algebra

## Similarity implies equality?

Let be similar to . Does hold?

Solution

No! Take then . The matrices are similar but not equal.

## On tensors

Let -Vect, , -linear. Prove that

Solution

Recall the general definition of the tensor product of linear maps, we have successively:

Thus, the two linear maps are equal when composed with the canonical bilinear map , hence equal (by the universal property).

## No solution

Let . Show that

has no solutions.

Solution

Since taking traces on both sides, we have

## Zero determinant

Consider the real numbers for . Prove that

Solution

Using the identity in combination with we have:

## Zero determinant

Let such that

Prove that if is odd.

Solution

Let . Then

Taking conjugate transpose we also have that

Hence . However it also holds . Combiming these two we get that

If we are done. Otherwise is real. In that case we have

since is odd. Hence as wanted.