How to calculate the Frobenius inner product of two matrices using Python NumPy?

by Amrita Mitra | Oct 3, 2023 | Featured, Linear Algebra

What is the Frobenius inner product of two matrices?

Let’s say A and B are two matrices with the same number of rows and columns. The Frobenius inner product of these two matrices will return a scalar such that the scalar is equal to the trace of the matrix A^TB.

${\langle A,B \rangle}_F=tr(A^TB)$

This calculation is very similar to vectorizing the two matrices and calculating their dot product. Please note that we can vectorize a matrix by concatenating all the columns of the matrix to form a single column.

For example, let’s say A and B are two matrices as given below.

$A={\begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}} \\ B={\begin{bmatrix} g & h & i \\ j & k & l \end{bmatrix}}$

So, if we vectorize A and B, we get the following:

$vect(A)={\begin{bmatrix} a \\ b \\ c \\ d \\ e \\ f \end{bmatrix}} \\ vect(B)={\begin{bmatrix} g \\ h \\ i \\ j \\ k \\ l \end{bmatrix}}$

So, the Frobenius inner product of A and B will be:

${\langle A,B \rangle}_F=vect(A) \cdot vect(B) \\ ={\begin{bmatrix} a \\ b \\ c \\ d \\ e \\ f \end{bmatrix}} \cdot {\begin{bmatrix} g \\ h \\ i \\ j \\ k \\ l \end{bmatrix}} \\ =ag+dj+bh+ek+ci+fl$

If we take a transpose of A and multiply A^T with B, we will get:

$A^TB={\begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}}^T{\begin{bmatrix} g & h & i \\ j & k & l \end{bmatrix}} \\ ={\begin{bmatrix} a & d \\ b & e \\ c & f \end{bmatrix}}{\begin{bmatrix} g & h & i \\ j & k & l \end{bmatrix}} \\ ={\begin{bmatrix} ag+dj & ah+dk & ai+dl \\ bg+ej & bh+ek & bi+el \\ cg+fj & ch+fk & ci+fl \end{bmatrix}} \\ tr(A^TB)=ag+dj+bh+ek+ci+fl$

So, we can calculate the Frobenius inner product of two matrices A and B in two ways. We can vectorize the matrices and calculate their dot product. We can also calculate the trace of the matrix A^TB.

…

Pages: 1 2

Calculate the pseudoinverse of a matrix using Python

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

What is the pseudoinverse of a matrix? We know that if A is a square matrix with full rank, then A-1 is said to be the inverse of A if the following condition holds: $latex AA^{-1}=A^{-1}A=I $ The pseudoinverse or the Moore-Penrose inverse of a matrix is a...

Cholesky decomposition using Python

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

What is Cholesky decomposition? A square matrix A is said to have Cholesky decomposition if it can be written as a product of a lower triangular matrix and its conjugate transpose. $latex A=LL^{*} $ If all the entries of A are real numbers, then the conjugate...

Tensor Hadamard Product using Python

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

In one of our previous articles, we already discussed what the Hadamard product in linear algebra is. We discussed that if A and B are two matrices of size mxn, then the Hadamard product of A and B is another mxn matrix C such that: $latex H_{i,j}=A_{i,j} \times...

Perform tensor addition and subtraction using Python

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

We can use numpy nd-array to create a tensor in Python. We can use the following Python code to perform tensor addition and subtraction. import numpy A = numpy.random.randint(low=1, high=10, size=(3, 3, 3)) B = numpy.random.randint(low=1, high=10, size=(3, 3, 3)) C =...

How to create a tensor using Python?

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

What is a tensor? A tensor is a generalization of vectors and matrices. It is easily understood as a multidimensional array. For example, in machine learning, we can organize data in an m-way array and refer it as a data tensor. Data related to images, sounds, movies,...

How to combine NumPy arrays using horizontal stack?

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

We can use the hstack() function from the numpy module to combine two or more NumPy arrays horizontally. For example, we can use the following Python code to combine three NumPy arrays horizontally. import numpy A = numpy.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) B =...

How to combine NumPy arrays using vertical stack?

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

Let’s say we have two or more NumPy arrays. We can combine these NumPy arrays vertically using the vstack() function from the numpy module. For example, we can use the following Python code to combine three NumPy arrays vertically. import numpy A = numpy.array([[1, 2,...

Singular Value Decomposition (SVD) using Python

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

What is Singular Value Decomposition (SVD)? Let A be an mxn rectangular matrix. Using Singular Value Decomposition (SVD), we can decompose the matrix A in the following way: $latex A_{m \times n}=U_{m \times m}S_{m \times n}V_{n \times n}^T $ Here, U is an mxm matrix....

Eigen decomposition of a square matrix using Python

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

Let A be a square matrix. Let’s say A has k eigenvalues λ1, λ2, ... λk. And the corresponding eigenvectors are X1, X2, ... Xk. $latex X_1=\begin{bmatrix} x_{11} \\ x_{21} \\ x_{31} \\ ... \\ x_{k1} \end{bmatrix} \\ X_2=\begin{bmatrix} x_{12} \\ x_{22} \\ x_{32} \\ ......

How to calculate eigenvalues and eigenvectors using Python?

by Amrita Mitra | October 3, 2023 | Featured, Linear Algebra | 0 Comments

In our previous article, we discussed what eigen values and eigenvectors of a square matrix are and how we can calculate the eigenvalues and eigenvectors of a square matrix mathematically. We discussed that if A is a square matrix, then $latex (A- \lambda I) \vec{u}=0...