Wednesday, 27 May 2020

Transpose Properties and Special Matrices

Transpose it very simple operation to change rows into columns and columns into rows. I had never given any importance to this operation until i found the relationship between Transpose Operation with Special Matrices. For Example we prefer Transpose of Matrix A over Inverse of Matrix A when we find the matrix is Orthogonal.

Orthogonal Matrix is a special matrix, I thought it is the only case. The real surprise came in, when dealing with Real Skew Symmetric Matrix.

In this blog I like to capture the properties of Transpose Operation of Matrix and the relation of Transpose operation with Special Matrices.

Properties of Transpose Operation: 
Let us see the properties

1. Inverse of Transpose: Transpose of Transpose
{\displaystyle \left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\mathbf {A} .}

 2. Transpose with Additivity
{\displaystyle \left(\mathbf {A} +\mathbf {B} \right)^{\operatorname {T} }=\mathbf {A} ^{\operatorname {T} }+\mathbf {B} ^{\operatorname {T} }.}

3. Transpose with Mulitiplicativity
{\displaystyle \left(\mathbf {AB} \right)^{\operatorname {T} }=\mathbf {B} ^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }.}

4. Transpose with Scalar Multiplication
{\displaystyle \left(c\mathbf {A} \right)^{\operatorname {T} }=c\mathbf {A} ^{\operatorname {T} }.}

5. Transpose with Determinant
{\displaystyle \det \left(\mathbf {A} ^{\operatorname {T} }\right)=\det(\mathbf {A} ).}

6. Share Same Eigen Values & Polynomial: 
AT and A share same eigen values as they polynomial representation is same.

7. Dot Product with Transpose
{\displaystyle \left[\mathbf {a} \cdot \mathbf {b} \right]=\mathbf {a} ^{\operatorname {T} }\mathbf {b} ,}

8. Transpose exchange with Inverse
{\displaystyle \left(\mathbf {A} ^{\operatorname {T} }\right)^{-1}=\left(\mathbf {A} ^{-1}\right)^{\operatorname {T} }.}

9. If A has all Real entries then
ATA is a positive-semidefinite matrix. Useful for defect and condition calculation to reduce error.

10. {\displaystyle \left(\mathbf {A} \mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }=\mathbf {A} \mathbf {A} ^{\operatorname {T} }.}
 

We have seen 10 properties of Transpose Operation. Now let us see What it has to do with Special Matrices. We shall start with simple special to very special and we shall see what special property arises with it.

If A is Symmetric Matrix, 
{\displaystyle \mathbf {A} ^{\operatorname {T} }=\mathbf {A} .} 
Only if A = AT we call a matrix Symmetric.


If A is Orthogonal Matrix, We call a matrix A orthogonal ATA = AAT = I
{\displaystyle \mathbf {A} ^{\operatorname {T} }=\mathbf {A} ^{-1}.}


{\displaystyle \mathbf {A} ^{\operatorname {T} }=-\mathbf {A} .}
Only if  AT = - A we call a matrix Skew Symmetric.


If A is Hermitian Matrix, We call a matrix H hermitian if H = Conjugate Transpose of H.
{\displaystyle \mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} }}.}
when we take transpose of hermitian we get AT = conjugate of A


{\displaystyle \mathbf {A} ^{\operatorname {T} }=-{\overline {\mathbf {A} }}.}
This is a combination of both Skewed Matix and Hermitian.

If A is Unitary Matrix, We call a matrix U unitary if U−1 = U∗ or U∗U = UU∗  = I. Here * denotes conjugate transpose.
{\displaystyle \mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} ^{-1}}}.}

For a unitary matrix U, determinant of U is 1. Hence it is called Unitary Matrix.

In the above cases we can see how various complex operation can be replaced with Transpose Operation, It is very interesting to know the application of Transpose :)

You can also read more about Special Matrices my other post.


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