## What are the properties of similar matrices?

If two matrices are similar, they have the same eigenvalues and the same number of independent eigenvectors (but probably not the same eigenvectors).

### What does Unitarily similar mean?

Unitarily similar means similarity by a unitary matrix, e.g. A=Q∗BQ where Q is unitary. The interest in unitary similarity is two-fold.

#### How do you know if a matrix is similar?

Two matrices A and B are similar if there exists a nonsingular (invertible) matrix S such […] If 2 by 2 Matrices Satisfy A=AB−BA, then A2 is Zero Matrix Let A,B be complex 2×2 matrices satisfying the relation A=AB−BA.

**How do you know if two 3×3 matrices are similar?**

Examine the properties of similar matrices. Do they have the same rank, the same trace, the same determinant, the same eigenvalues, the same characteristic polynomial. If any of these are different then the matrices are not similar.

**What are two similar matrices?**

Two square matrices are said to be similar if they represent the same linear operator under different bases. Two similar matrices have the same rank, trace, determinant and eigenvalues.

## Why do similar matrices have the same determinant?

Square matrices A and B of the same order related by B=S−1AS, where S is a non-singular matrix of the same order. Similar matrices have the same rank, the same determinant, the same characteristic polynomial, and the same eigenvalues.

### How do you know if two matrices are unitarily equivalent?

Two matrices A, B ∈ Mn are unitarily equivalent if ∃ U ∈ Mn such that U∗U = In and B = U∗A U.

#### Why do similar matrices have the same trace?

**Do similar matrices have the same singular values?**

Clearly not. E.g. {A(x)=(0x00): x≠0} is a family of similar matrices, but the singular values of A(x) are |x| and zero.

**Do similar matrices have the same determinant?**

Similar matrices have the same rank, the same determinant, the same characteristic polynomial, and the same eigenvalues.

## Does every matrix have a similar matrix?

For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically closed field), every matrix is similar to a matrix in Jordan form.

### How do you show similar matrices that have the same determinant?

Suppose that A and B are similar, i.e. that B = P–1AP for some matrix P. so the matrices have the same determinant, and one is invertible if the other is. so the matrices have the same characteristic polynomial and hence the same eigenvalues. Note that similar matrices will not generally have the same eigenvectors.

#### When are two matrices said to be unitarily similar?

In linear algebra we often use the term “unitarily similar”. Definition Two matrices and are said to be unitarily similar if and only if there exists a unitary matrix such that Thus, two matrices are unitarily similar if they are similar and their change-of-basis matrix is unitary.

**What are the properties of a similar matrix?**

Proposition Matrix similarity is an equivalence relation, that is, given three matrices , and , the following properties hold: Reflexivity: is similar to itself; Symmetry: if is similar to , then is similar to ; Transitivity: if is similar to and is similar to , then is similar to .

**When is a matrix A conjugate unitary matrix?**

A is conjugate unitary if every matrix unitarily similar to A, is conjugate unitary. Proof. Assume that A is conjugate unitary ) AA = A A = I. Let B is any matrix, which is unitarily similar to A. Therefore B = U AU where U is any unitary matrix.

## When are two square matrices have the same rank?

Two square matrices are said to be similar if they represent the same linear operator under different bases. Two similar matrices have the same rank, trace, determinant and eigenvalues.