# Do similar matrices have same singular values?

## Do similar matrices have 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.

**Can similar matrices be singular?**

No, they do not. Let A be an mxn matrix; denote the eigenvalues of a square matrix X by \lambda_{j} (X), and denote the singular values of a matrix Y by s_{j} (Y). The second equality in (3) follows from the fact that matrix similarity preserves the eigenvalues of the matrix.

**Are similar matrix have same?**

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.

### Do two similar matrices have the same eigenvalues?

If two matrices are similar, they have the same eigenvalues and the same number of independent eigenvectors (but probably not the same eigenvectors). When we diagonalize A, we’re finding a diagonal matrix A that is similar to A.

**What is meant by similar matrices?**

Definition (Similar Matrices) Suppose A and B are two square matrices of size n . Then A and B are similar if there exists a nonsingular matrix of size n , S , such that A=S−1BS A = S − 1 B S .

**How do you find similar matrices?**

Also, if two matrices have the same distinct eigen values then they are similar. Suppose A and B have the same distinct eigenvalues. Then they are both diagonalizable with the same diagonal 2 Page 3 matrix A. So, both A and B are similar to A, and therefore A is similar to B.

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

More generally, we can prove that if A and B are similar, then their characteristic polynomials are the same. From this, we also can deduce that the determinants of A and B are the same as well as their traces are the same. For a proof, see the post “Similar matrices have the same eigenvalues“.

**Do similar matrices have the same eigenvalues with the same multiplicities?**

Of course: similar matrices have the same characteristic polynomial. Hence, the eigenvalues have the same algebraic multiplicities.

**How do you know when matrices are similar?**

## What does it mean if two matrices are similar?

In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that. Similar matrices represent the same linear map under two (possibly) different bases, with P being the change of basis matrix.

**Why similar matrices have the same eigenvalues?**

Since similar matrices A and B have the same characteristic polynomial, they also have the same eigenvalues. If B = PAP−1 and v = 0 is an eigenvector of A (say Av = λv) then B(Pv) = PAP−1(Pv) = PA(P−1P)v = PAv = λPv. Thus Pv (which is non-zero since P is invertible) is an eigenvector for B with eigenvalue λ.

**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.