More on Matrices

Learn how we can perform some advanced matrix operations using JAX.

We'll cover the following

While the vast world of matrices and the implementation is more than even a dedicated course can cover, we’ll try to do some justice to the subject by concluding with another lesson covering a few important concepts.

Determinant

Like the norm, the determinant is a scalar value characterizing a square matrix. It is quite useful in defining some properties of the matrix.

For example, a matrix is singular if (and only if) its determinant is zero.

We can calculate it as:

a = linalg.det(A)

Inverse

The inverse of a matrix is defined as:

A−1=Adj.A∣A∣A^{-1} = \frac{Adj. A}{|A|}

The linalg sublibrary provides the function for inverse calculation as:

A_inv = linalg.inv(A)

Note: The linalg.pinv() can be used to calculate a pseudo-inverseWhile inverse is calculated only for square, full-rank matrices, Pseudo/Moore-Penrose inverse can be calculated for any mxn matrix. and doesn’t support the int members.

Power

Similarly, we can calculate the power of A using matrix_power() as:

A_square = linalg.matrix_power(A,2)
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A = jnp.array([[1, 2, 3],[ 4, 5, 6],[7, 8, 9]])
print(A)
B = linalg.matrix_power(A,-1)
C = linalg.inv(A)
d = linalg.det(A)
print(B)
print(C) # will have same answer as B
print("Matrix's determinant is: ",d)

A is singularA matrix having 0 determinant and hence an undefined inverse., so the outputs above should not be a surprise. Below is a non-singular matrix with its square and inverse respectively:

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A = jnp.array([[1, -1, 3],[ 3, 5, 1],[7, 0, 9]])
print(A)
B = linalg.matrix_power(A,2)
C = linalg.inv(A)
D = linalg.inv(C) #Allowing rounding-off, D=A
e = linalg.det(A)
print("Matrix's determinant is: ",e)
print(B)
print(C)
print(D)

Eigenvectors and eigenvalues

If we recall, an eigenvector v of a square matrix A is defined as:

Av=λvAv=\lambda v

where λ is the corresponding eigenvalue.

lamba_values, eigen_vect = linalg.eig(A)

Compared to normal NumPy, the JAX version treats every value as complex, so don’t be surprised by the j we observe in the output values.

It may be tempting to use “lambda” as a variable identifier. Don’t do this since lambda is a keyword. Instead, we can use λ itself as an identifier.

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A = jnp.array([[2,1,5],[3,0,4],[2,1,-2]])
λ, eigen_vectors = linalg.eig(A)
print("The λ values are:,",λ)
print("And Eigen vectors (as a matrix) are:",eigen_vectors)

Definiteness

Eigenvalues of a vector introduce another related concept.

We call a square matrix a positive definite matrix if all the eigenvalues are positive while a positive semidefinite matrix is one having all its eigenvalues either zero or positive.

We can extend this concept to negative definite and semidefinite matrices as well.

If a matrix has some positive eigenvalues and other negatives, it’s called an indefinite matrix. These matrices are pretty helpful as we will shortly see in the follow-up lessons.

Note: The concept of definiteness is usually restricted to only real-value square matrices, but can be expanded to complex matrices if needed.

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def TestDefiniteness(X):
  if(jnp.all(linalg.eigvals(X) > 0)):
    print("It's Positive Definite")
  elif(jnp.all(linalg.eigvals(X) >= 0)):
    print("It's Positive Semidefinite")
  elif(jnp.all(linalg.eigvals(X) < 0)):
    print("It's Negative Definite")
  elif(jnp.all(linalg.eigvals(X) <= 0)):
    print("It's Negative Semidefinite")
  else:
    print("Its Indefinite")
#Lets test some matrices
A = jnp.array(([2.0,3.0,4.0],[1.2,2.4,5.2],[2.8,3.6,4.9]))
B = jnp.array(([2.4,4.3],[6.1, 13.6]))
TestDefiniteness(A)
TestDefiniteness(B)

Singular Value Decomposition

We’ll conclude the lesson by performing a singular value decomposition (SVD) on a given matrix. If you recall, an SVD of matrix A is:

U Σ V∗=AU~\Sigma ~V^* = A

If A is m×nm\times n, then the dimensions of the decomposed matrices are:

U:m×mU: m \times m

Σ:m×n\Sigma: m \times n

V∗:n×nV^*: n \times n

We can verify the dimensions in this example:

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A = jnp.array([[2,1,5],[3,0,4],[2,1,-2],[4,0,7]])
U, Sigma, V = linalg.svd(A)
print("Original matrix shape:",A.shape)
print("U shape:",U.shape)
print("Σ shape:",Sigma.shape)
print("V* shape:", V.shape)

If we look closely, while the shapes of UU and V∗V^* are consistent with the formula, Σ\Sigma is returned as a one-dimensional vector of singular values, which is consistent with NumPy.